Spreadsheets — GEOMETOR • fundamentals

Workbooks in Archive¶

resources/spreadsheets/Triangles and Circles.ods — Master workbook containing 59 worksheets spanning triangle centers, circle families, harmonic conjugates, projective theorems, and special triangles.
resources/spreadsheets/00.triangle-basics/circumcircle-circumcenter.ods — Step-by-step derivation of the circumcenter and circumcircle from side lengths $(5, 6, 7)$ (also exported as .csv, .html, .fods, and .xlsx).
resources/spreadsheets/00.triangle-basics/triangle-basics-draw.ods — Coordinate plotting workbook for basic triangle vertices and centers.

Master Worksheet Index (`Triangles and Circles.ods`)¶

#	Worksheet	Active Rows	Linked Codex Topics
1	`DrawTriangle`	44	General Reference
2	`Sides`	42	General Reference
3	`Basic Centers`	0	General Reference
4	`triangle centered at incenter`	43	General Reference
5	`3pts=1circle`	43	General Reference
6	`Sides - centroid`	45	General Reference
7	`Incircle`	42	General Reference
8	`incircle 2 cot`	44	General Reference
9	`Circumcircle`	44	General Reference
10	`Circumcircle 2`	43	General Reference
11	`Euler`	41	General Reference
12	`dIO`	40	General Reference
13	`centroid`	32	General Reference
14	`geo mean`	37	General Reference
15	`Orthocenter`	44	F.34, F.36, F.50
16	`Euler 2x 3x angle idents`	39	General Reference
17	`Excircles`	43	General Reference
18	`Extangents`	42	General Reference
19	`Orthic Triangle`	43	F.36, F.38
20	`Fermat`	44	General Reference
21	`Brocard`	43	General Reference
22	`Arc of C`	8	General Reference
23	`Pythagorean`	45	General Reference
24	`PT w equal hyp`	43	General Reference
25	`PythagoreanTripleGraph`	34	General Reference
26	`PythTrianglewExcircles`	39	General Reference
27	`PythTriApexParameter`	45	General Reference
28	`HeronianTriangle`	25	General Reference
29	`Angle Bisector Thm`	44	General Reference
30	`Alt Segments Thm`	16	General Reference
31	`Menelaus`	40	F.7
32	`Ceva`	37	F.7
33	`Altitude`	33	General Reference
34	`Point in Triangle Test`	42	General Reference
35	`Point & Triangle`	44	General Reference
36	`barycentric`	41	General Reference
37	`barycentric w CaR`	37	General Reference
38	`Trilinear`	41	General Reference
39	`trace`	44	General Reference
40	`Power`	21	General Reference
41	`Intersecting Circles`	36	General Reference
42	`Isotomic Conj`	17	General Reference
43	`Harmonic Conj`	40	General Reference
44	`HarmConj`	39	General Reference
45	`HC XYZ`	15	General Reference
46	`HarmMean`	37	General Reference
47	`Orthogonal Circles`	44	General Reference
48	`Orthogonal 2`	44	General Reference
49	`Isosceles`	42	General Reference
50	`Golden Triangle`	42	F.23, F.40
51	`Equil w Known Radials`	23	General Reference
52	`Oai Equil ∆`	41	General Reference
53	`20-40-120`	43	General Reference
54	`Euler 2x 3x angles`	34	General Reference
55	`Special`	42	General Reference
56	`Hyperbolas of ∆`	18	General Reference
57	`Grk Sym`	44	General Reference
58	`E2 ASCII`	23	General Reference
59	`circle templ`	43	General Reference

Worksheet Formula Inspector¶

Expand any worksheet below to inspect its OpenFormula expressions and computed values.

Worksheet: DrawTriangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G9`	G9	`4`	`=D9`
`H9`	4	`0`	`=E9`
`G10`	G10	`0`	`=D10`
`H10`	0	`3`	`=E10`
`G11`	G11	`0`	`=D11`
`H11`	0	`0`	`=E11`
`C12`	C12	`A`	`=C9`
`D12`	A	`4`	`=D9`

Cell Excerpt

Triangle Basics	—	—	—	—	—
The semiperimeter s is equal to (a+b	—	—	—	—	—
The area K is equal to the square ro	—	—	—	—	—
Efimov trimer dimensions (Å)	—	—	—	—	—
Fred Seelig	—	—	—	—	—
May 25, 2011	—	—	—	—	—

Worksheet: Sides (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G16`	G16	`a`	`=G16`
`H16`	a	`3`	`=H16`
`G17`	G17	`b`	`=G17`
`H17`	b	`3`	`=H17`
`G18`	G18	`c`	`=G18`
`H18`	c	`5`	`=H18`
`H20`	A	`0.0`	`=-[.$H$12]/2`
`H21`	B	`5.0`	`=H18`

Cell Excerpt

Triangle Basics	—	—	—	—	—
Given the side lengths, determine an	—	—	—	—	—
The semiperimeter s is equal to (a+b	—	—	—	—	—
The area K is equal to the square ro	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Oct. 28, 2011	—	—	—	—	—

Worksheet: Basic Centers (Triangles and Circles.ods)

Auxiliary layout or scratch worksheet in workbook.

Worksheet: triangle centered at incenter (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J11`	A	`-5.0`	`=-G25`
`K11`	-5.0	`-2.8`	`=-H25`
`J12`	B	`6.0`	`=G13-G25`
`K12`	6.0	`-2.8`	`=-H25`
`J13`	C	`-0.4`	`=G12*COS(H17)-G25`
`K13`	-0.4	`4.9`	`=G12*SIN(H17)-H25`
`F14`	F14	`a`	`=F11`
`G14`	a	`10`	`=G11`

Cell Excerpt

A Triangle Whose Incenter is the Ori	—	—	—	—	—
Given the side lengths, determine an	—	—	—	—	—
The triangle will be centered on the	—	—	—	—	—
The semiperimeter s is equal to (a+b	—	—	—	—	—
The area K is equal to the square ro	—	—	—	—	—
Fred Seelig	—	—	—	—	—

Worksheet: 3pts=1circle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`E14`	A	`-2`	`=E12`
`F14`	-2	`-2`	`=F12`
`E15`	B	`5`	`=E13`
`F15`	5	`3`	`=F13`
`E16`	dA	`8.0`	`=E12^2+F12^2`
`F16`	8.0	`8.0`	`=E17`
`E17`	dB	`34.0`	`=E13^2+F13^2`
`F17`	34.0	`34.0`	`=E18`

Cell Excerpt

Three Points Specify One Circle Uniq	—	—	—	—	—
Define three points	—	—	—	—	—
This gives a compact, modular way of	—	—	—	—	—
Graphical points are included too	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Oct. 31, 2011	—	—	—	—	—

Worksheet: Sides - centroid (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`E12`	E12	`7.61577310586391`	`=SQRT(58)`
`E13`	E13	`3.60555127546399`	`=SQRT(13)`
`C15`	(hide)	`a`	`=C12`
`E15`	E15	`7.61577310586391`	`=E12`
`C16`	(hide)	`b`	`=C13`
`E16`	E16	`3.60555127546399`	`=E13`
`E17`	E17	`5`	`=E14`
`E19`	E19	`123.7`	`=DEGREES(F19)`

Cell Excerpt

Triangle Basics	—	—	—	—	—
Given the side lengths, determine an	—	—	—	—	—
The triangle will be graphed with it	—	—	—	—	—
The triangle will be rotatable by a	—	—	—	—	—
The semiperimeter s is equal to (a+b	—	—	—	—	—
The area K is equal to the square ro	—	—	—	—	—

Worksheet: Incircle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`K11`	K11	`-2`	`=E12`
`L11`	-2	`-2`	`=F12`
`K12`	K12	`5`	`=E13`
`L12`	5	`3`	`=F13`
`K13`	K13	`1`	`=E14`
`L13`	1	`4`	`=F14`
`K14`	K14	`-2`	`=E12`
`L14`	-2	`-2`	`=F12`

Cell Excerpt

Incircle / incenter	—	—	—	—	—
The incenter is the intersection of	—	—	—	—	—
The contact points form another tria	—	—	—	—	—
Fred Seelig	—	—	—	—	—
May 25, 2011	—	—	—	—	—
Updated Nov 27, 2011	—	—	—	—	—

Worksheet: incircle 2 cot (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`H11`	1	`1`	`=G11`
`G12`	r	`1`	`=G11`
`H12`	1	`1`	`=G12`
`G13`	r	`1`	`=G12`
`H13`	1	`1`	`=G13`
`G14`	r	`1`	`=G13`
`H14`	1	`1`	`=G14`
`G15`	r	`1`	`=G14`

Cell Excerpt

Incircle - a Triangle defined by the	—	—	—	—	—
A triangle can be specified by the r	—	—	—	—	—
the triangle’s three nterior angles	—	—	—	—	—
Side c is parallel to the x axis	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Created July 18, 2013	—	—	—	—	—

Worksheet: Circumcircle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G11`	-2	`13`	`=E12^2+F12^2`
`J11`	J11	`-3`	`=E12`
`K11`	-3	`-2`	`=F12`
`G12`	0	`36`	`=E13^2+F13^2`
`J12`	J12	`6`	`=E13`
`K12`	6	`0`	`=F13`
`G13`	7	`53`	`=E14^2+F14^2`
`J13`	J13	`2`	`=E14`

Cell Excerpt

Circumcircle / Circumcenter	—	—	—	—	—
The circumcenter is the intersection	—	—	—	—	—
It is the intersection of the perpen	—	—	—	—	—
This can be used to fit a circle to	—	—	—	—	—
Fred Seelig	—	—	—	—	—
May 25, 2011	—	—	—	—	—

Worksheet: Circumcircle 2 (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`H11`	1	`1`	`=G11`
`G12`	R	`1`	`=G11`
`H12`	1	`1`	`=G12`
`G13`	R	`1`	`=G12`
`H13`	1	`1`	`=G13`
`G14`	R	`1`	`=G13`
`H14`	1	`1`	`=G14`
`G15`	R	`1`	`=G14`

Cell Excerpt

Circumcircle / Circumcenter	—	—	—	—	—
The circumcenter is the origin of a	—	—	—	—	—
The three vertices are specified by	—	—	—	—	—
Side c is parallel to the x axis	—	—	—	—	—
Fred Seelig	—	—	—	—	—
July 17 2013	—	—	—	—	—

Worksheet: Euler (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G21`	s	`1.5`	`=SUM(G20:G22)/2`
`G22`	K	`0.4`	`=SQRT(G23(G23-G20)(G23-G21)*(G23-G22))`
`G23`	R	`0.6`	`=PRODUCT(G20:G22)/(4*G24)`
`H23`	0.6	`0.6`	`=G25`
`I24`	I24	`360`	`=SUM(I148:I150)`
`G25`	fR	`3.7`	`=(3*PI()/2)-(G150/2)`
`I25`	I25	`210`	`=DEGREES(G27)`
`G27`	qA	`3.7`	`=[.G$27]`

Cell Excerpt

The Euler Line

The Euler line L is the line segment

where circumcenter-centroid distance

Incenter I is only on this line for

All four points merge into one point

Fred Seelig

Worksheet: dIO (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`H13`	a	`1.61803`	`=(1+SQRT(5))/2`
`H14`	b	`1.61803`	`=(1+SQRT(5))/2`
`G16`	G16	`a`	`=G16`
`H16`	a	`1.6180339887499`	`=H16`
`G17`	G17	`b`	`=G17`
`H17`	b	`1.6180339887499`	`=H17`
`G18`	G18	`c`	`=G18`
`H18`	c	`1`	`=H18`

Cell Excerpt

dIO	—	—	—	—	—
Given the reference triangle’s side	—	—	—	—	—
dIO2 = R2 - 2rR	—	—	—	—	—
Fred Seelig	—	—	—	—	—
May 22, 2014	—	Created	—	—	—
May 22, 2014	—	—	Last Update	—	—

Worksheet: centroid (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`I12`	I12	`-2`	`=D12`
`J12`	-2	`-4`	`=E12`
`L12`	L12	`-2`	`=D12`
`I13`	I13	`2`	`=D13`
`J13`	2	`3`	`=E13`
`L13`	L13	`4.5`	`=D19`
`I14`	I14	`7`	`=D14`
`J14`	7	`1`	`=E14`

Cell Excerpt

Centroid, or Orthocenter	—	—	—	—	—
The centroid of a triangle is the in	—	—	—	—	—
A median is the line segment that co	—	—	—	—	—
The centroid is two thirds the lengt	—	—	—	—	—
Fred Seelig	—	—	—	—	—
June 1, 2011	—	—	—	—	—

Worksheet: geo mean (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F23`	y	`1.6`	`=F21-F22`
`F24`	g	`0.8`	`=SQRT(F22*F23)`
`F33`	x	`1.6`	`=(F31/2)(1+SQRT(1-(2F32/F31)^2))`
`F34`	y	`0.4`	`=(F31/2)(1-SQRT(1-(2F32/F31)^2))`
`F40`	F40	`3.64`	`=F38^2-4*F39^2`
`F42`	F42	`0.91`	`=1-(2*F39/F38)^2`

Cell Excerpt

Geometric Mean	—	—	—
(Based on Thales theorem)	—	—	—
Fred W. Seelig	—	—	—
Feb. 27, 2014	—	—	Created
Feb. 27, 2014	—	—	Last Update
This worksheet shows the relationshi	—	—	—

Worksheet: Orthocenter (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`I5`	I5	`A`	`=D12`
`J5`	A	`0`	`=E12`
`K5`	0	`0`	`=F12`
`H6`	H6	`B`	`=D13`
`I6`	B	`6`	`=E13`
`J6`	6	`0`	`=F13`
`I7`	I7	`C`	`=D14`
`J7`	C	`-2`	`=E14`

Cell Excerpt

Orthocenter	—	—	—	—	—
The orthocenter H is the intersecti	—	—	—	—	—
When the reference triangle is obtus	—	—	—	—	—
and has radius rPOLAR2 = 4R2 - (a2+b	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Begun 6/1/2011 and completely rework	—	—	—	—	—

Worksheet: Euler 2x 3x angle idents (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G23`	15	`0.261799387799149`	`=RADIANS(F23)`
`K23`	15	`0.261799387799149`	`=RADIANS(J23)`
`F24`	2x	`30`	`=2*F23`
`G24`	30	`0.523598775598299`	`=RADIANS(F24)`
`J24`	3x	`45`	`=3*J23`
`K24`	45	`0.785398163397448`	`=RADIANS(J24)`
`F25`	180-3x	`135`	`=180-SUM(F23:F24)`
`G25`	135	`2.35619449019234`	`=RADIANS(F25)`

Cell Excerpt

Euler Double and Triple Angle Formul

Given a, and b = 2a, then g = 180°-

If c is given, then by law of sines,

Given a, and b = 3a, then g = 180°-

If c is given, then by law of sines,

For the special case when the inradi

Worksheet: Excircles (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F19`	A	`-4`	`=F16`
`G19`	-4	`-2`	`=G16`
`F20`	B	`2`	`=F17`
`G20`	2	`-1`	`=G17`
`F21`	C	`-1`	`=F18`
`G21`	-1	`5`	`=G18`
`F23`	a	`6.7`	`=SQRT((F18-F17)^2+(G18-G17)^2)`
`G23`	6.7	`6.7`	`=F23`

Cell Excerpt

Excircles

Each triangle has three excircles, A

The lines connecting the excircle ce

I do not know the name of the point

Create a line connecting a vertex to

They all meet at the same point, cal

Worksheet: Extangents (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F16`	A	`-0.8205`	`=-F26/2`
`F17`	B	`0.8205`	`=-F16`
`F18`	C	`-1.0133175`	`=F26*F23/2`
`G18`	-1.0133175	`1.2187690432663`	`=2/F26`
`F19`	A	`-0.8205`	`=F16`
`G19`	-0.8205	`0`	`=G16`
`F20`	B	`0.8205`	`=F17`
`G20`	0.8205	`0`	`=G17`

Cell Excerpt

Extangents Triangle

Extangents triangle T

This investigates the side lengths a

The reference triangle is unity area

Side length c is a free parameter, a

t is the normalization parameter tha

Worksheet: Orthic Triangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G18`	G18	`5`	`=G16`
`H18`	5	`5`	`=H16`
`G19`	G19	`-9`	`=G17`
`H19`	-9	`4`	`=H17`
`G20`	G20	`0`	`=G18`
`H20`	0	`6`	`=H18`
`G22`	G22	`9.2`	`=SQRT((G18-G17)^2+(H18-H17)^2)`
`H22`	9.2	`9.2`	`=G23`

Cell Excerpt

Orthic Triangle	—	—	—	—	—
The reference triangle DR’s orthic t	—	—	—	—	—
The three vertices are the three foo	—	—	—	—	—
The orthic center H is the incenter	—	—	—	—	—
This is an incredibly useful triangl	—	—	—	—	—
Since most sketches start with a DC,	—	—	—	—	—

Worksheet: Fermat (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`L9`	L9	`2.8`	`=D11`
`D11`	F	`2.8`	`=D35`
`E11`	2.8	`1.3`	`=E35`
`I12`	I12	`0`	`=D13`
`J12`	0	`0`	`=E13`
`L12`	L12	`0`	`=D13`
`I13`	I13	`3`	`=D14`
`J13`	3	`4`	`=E14`

Cell Excerpt

Fermat Point	—	—	—	—	—
The sum of distances from all three	—	—	—	—	—
If the triangle is acute, F is insid	—	—	—	—	—
If the triangle is obtuse, F is equa	—	—	—	—	—
The angles at the Fermat point shoul	—	—	—	—	—
Fred Seelig	—	—	—	—	—

Worksheet: Brocard (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G14`	G14	`a`	`=G16`
`H14`	a	`5`	`=H16`
`G15`	G15	`b`	`=G17`
`H15`	b	`3`	`=H17`
`G16`	G16	`c`	`=G18`
`H16`	c	`4`	`=H18`
`H18`	A	`0.0`	`=-[.$H$12]/2`
`H19`	B	`4.0`	`=H18`

Cell Excerpt

Brocard Points	—	—	—	—	—
David Wells, “Curious and Interestin	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Oct. 28, 2011	—	—	—	—	—
August 28, 2014	—	—	—	—	—
STATUS:	—	—	—	—	—

Worksheet: Arc of C (Triangles and Circles.ods)

Cell Excerpt

The Arc of C	—	—	—
Triangle ABC, with fixed C and const	—	—	—
Investigating the equation for area	—	—	—
or of a parameter angle q, where	—	—	—
Companion C.a.R. construction	—	—	—
“Arc of C.zir”	—	—	—

Worksheet: Pythagorean (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`B16`	1	`1`	`=GCD(D16;E16)`
`C16`	1	`1`	`=GCD(F16;G16;H16)`
`F16`	1	`3`	`=D16^2-E16^2`
`G16`	3	`4`	`=2D16E16`
`H16`	4	`5`	`=D16^2+E16^2`
`I16`	5	`3`	`=F16`
`J16`	3	`4`	`=G16`
`K16`	4	`5`	`=H16`

Cell Excerpt

Pythagorean Triple

A triangle having integer sidelength

With hypotenuse equal to the sqrt of

A triangle with PT sides always has

Fred Seelig

July 2, 2012

Worksheet: PT w equal hyp (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`B15`	1	`1`	`=GCD(D16;E16)`
`C15`	1	`1`	`=GCD(F16;G16;H16)`
`F15`	4	`33`	`=D16^2-E16^2`
`G15`	33	`56`	`=2D16E16`
`H15`	56	`65`	`=D16^2+E16^2`
`I15`	65	`33`	`=F16`
`J15`	33	`56`	`=G16`
`K15`	56	`65`	`=H16`

Cell Excerpt

Pythagorean Triangles w Same Hypoten	—	—	—	—	—
Useful when investigating cevians an	—	—	—	—	—
Fred W Seelig	—	—	—	—	—
July 20, 2013	—	—	—	—	Begun
Sept 4, 2014	—	—	—	—	Last Edit
M,N entries done by hand - very frus	—	—	—	—	—

Worksheet: PythagoreanTripleGraph (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F12`	1	`3`	`=D12^2-E12^2`
`G12`	3	`4`	`=2D12E12`
`H12`	4	`5`	`=D12^2+E12^2`
`I12`	5	`6`	`=(F12+G12+H12)/2`
`J12`	6	`1`	`=K12/I12`
`K12`	1	`6`	`=SQRT(I12(I12-F12)(I12-G12)*(I12-H12))`
`F13`	2	`5`	`=D13^2-E13^2`
`G13`	5	`12`	`=2D13E13`

Cell Excerpt

Pythagorean Triple	—	—	—	—	—
A triangle having integer sidelength	—	—	—	—	—
With hypotenuse equal to the sqrt of	—	—	—	—	—
A triangle with PT sides always has	—	—	—	—	—
Fred Seelig	—	—	—	—	—
1-Jun-11	—	—	—	—	—

Worksheet: PythTrianglewExcircles (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F17`	B	`5`	`=F26`
`G18`	0	`4`	`=F25`
`F19`	A	`0`	`=F17`
`G19`	0	`0`	`=G17`
`F20`	B	`5`	`=F18`
`G20`	5	`0`	`=G18`
`F21`	C	`0`	`=F19`
`G21`	0	`4`	`=G19`

Cell Excerpt

Pythagorean Triple Triangle with Exc	—	—	—	—	—
Extangent lines, and using points of	—	—	—	—	—
Fred W. Seelig Begun June 4, 2012	—	—	—	—	—
Last Edited June 4, 2012	—	—	—	—	—
Centers of Similitude equations foun	—	—	—	—	—
“Tangents to Two Circles.xls”	—	—	—	—	—

Worksheet: PythTriApexParameter (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`E15`	1	`3`	`=C15^2-D15^2`
`F15`	3	`4`	`=2C15D15`
`G15`	4	`5`	`=C15^2+D15^2`
`H15`	5	`6`	`=(E15+F15+G15)/2`
`I15`	6	`1`	`=J15/H15`
`J15`	1	`6`	`=SQRT(H15(H15-E15)(H15-F15)*(H15-G15))`
`L15`	L15	`NO`	`=IF(ABS(M15)<[.L$12];"YES";"NO")`
`B16`	B16	`2`	`=B15+1`

Cell Excerpt

Pythagorean Triple Apex Parameter	—	—	—	—	—
Let the long side of the pythagorean	—	—	—	—	—
having length c. Then the sides of l	—	—	—	—	—
This spreadsheet shows the apex offs	—	—	—	—	—
Fred Seelig	—	—	—	—	—
July 2, 2012	—	—	—	—	—

Worksheet: HeronianTriangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F18`	1	`0.894427190999916`	`=SQRT((D18^2E18)/(2D18+E18))`
`H18`	1	`1.4142135623731`	`=SQRT(D18*E18)`
`I18`	1.4142135623731	`5`	`=E18*(D18^2+G18^2)`
`J18`	5	`4`	`=D18*(E18^2+G18^2)`
`K18`	4	`3`	`=(D18+E18)(D18E18-G18^2)`
`L18`	3	`90.0˚`	`=DEGREES(ACOS((J18^2+K18^2-I18^2)/(2J18K18)))`
`F19`	1	`1.50755672288882`	`=SQRT((D19^2E19)/(2D19+E19))`
`H19`	2	`2.23606797749979`	`=SQRT(D19*E19)`

Cell Excerpt

Heronian Triples	—	—	—
A Heronian triangle has integer valu	—	—	—
Such a triangle can be created by ma	—	—	—
e.g., (5, 12, 13) with (12, 35, 37).	—	—	—
Some Heronian triangles exist that c	—	—	—
This does not decompose HTs into com	—	—	—

Worksheet: Angle Bisector Thm (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`C15`	(hide)	`a`	`=C12`
`E15`	E15	`28.6`	`=E12`
`C16`	(hide)	`b`	`=C13`
`E16`	E16	`9`	`=E13`
`E17`	E17	`30.8`	`=E14`
`J18`	g	`95.5`	`=E21`
`E19`	E19	`67.6`	`=DEGREES(F19)`
`F19`	67.6	`1.2`	`=ACOS(((E13^2+E14^2)-(E12^2))/(2E13E14))`

Cell Excerpt

Angle Bisector Theorem	—	—	—	—	—
Given the side lengths, determine an	—	—	—	—	—
The triangle will be graphed with it	—	—	—	—	—
The triangle will be rotatable by a	—	—	—	—	—
The semiperimeter s is equal to (a+b	—	—	—	—	—
The area K is equal to the square ro	—	—	—	—	—

Worksheet: Alt Segments Thm (Triangles and Circles.ods)

Cell Excerpt

Alternate Segments Theorem	—	—	—	—	—
—	Angles that subtend arcs	—	—	—	—
Euclid, Book III, Prop. 32	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Sept. 10, 2012	—	—	—	—	—
updated Sept 10, 2012	—	—	—	—	—

Worksheet: Menelaus (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J10`	J10	`5`	`=G10`
`K10`	5	`6`	`=H10`
`J11`	J11	`0`	`=G11`
`K11`	0	`0`	`=H11`
`J12`	J12	`18`	`=G12`
`K12`	18	`0`	`=H12`
`D13`	D13	`A`	`=D10`
`G13`	G13	`5`	`=G10`

Cell Excerpt

Menelaus Theorem	—	—	—	—	—
A line cuts the triangle ABC at thre	—	—	—	—	—
The product of ratios (BD/DC)*(CE/EA	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Begun Oct. 28, 2011	—	—	—	—	—
July 5, 2012	—	—	—	—	—

Worksheet: Ceva (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F33`	A	`-8`	`=-F25`
`I33`	I33	`-8`	`=F38`
`J33`	-8	`0`	`=G38`
`F34`	B	`16`	`=F27`
`I34`	I34	`16`	`=F39`
`J34`	16	`0`	`=G39`
`G35`	0	`8`	`=F26`
`I35`	I35	`0`	`=F40`

Cell Excerpt

Ceva’s Theorem	—
If three lines meet inside a triangl	—
Triangle ABC. Create three points D,	—
for all possible triangles, then K i	—
The product of side ratios equals un	—
(AF/BF)(BD/CD)(CE/AE) = 1	—

Worksheet: Altitude (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`E12`	E12	`3.67058823529412`	`=E10+((((E12-E10)(E11-E10))+((F12-F10)(F11-F10)))/((E11-E10)^2+(F11-F10)^2))*(`
`F12`	3.67058823529412	`-0.517647058823529`	`=F10+((((E12-E10)(E11-E10))+((F12-F10)(F11-F10)))/((E11-E10)^2+(F11-F10)^2))*(`
`E13`	E13	`7.7`	`=SQRT((E12-E13)^2+(F12-F13)^2)`
`F13`	7.7	`0.741176470588235`	`=(E24+F24)/E25`
`E23`	E23	`45`	`=(E23-E21)*(E22-E21)`
`F23`	45	`18`	`=(F23-F21)*(F22-F21)`
`E24`	E24	`85`	`=(E22-E21)^2+(F22-F21)^2`
`F24`	85	`85`	`=(E22-E21)^2+(F22-F21)^2`

Cell Excerpt

Finding the altitude from a point to	—	—	—	—	—
and finding the distance between a p	—	—	—	—	—
Fred Seelig	—	—	—	—	—
November 4, 2011	—	—	—	—	—
This is a portable, highly condensed	—	—	—	—	—
It is not coupled to the graph	—	—	—	x	y

Worksheet: Point in Triangle Test (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`K10`	K10	`0`	`=E10`
`L10`	0	`0`	`=F10`
`K11`	K11	`18`	`=E11`
`L11`	18	`0`	`=F11`
`K12`	K12	`6`	`=E12`
`L12`	6	`6`	`=F12`
`C13`	C13	`A`	`=C10`
`E13`	E13	`0`	`=E10`

Cell Excerpt

Point Inside, Outside, or On Triangl	—	—	—	—	—
A line cuts the triangle ABC at thre	—	—	—	—	—
The product of ratios (BD/DC)*(CE/EA	—	—	—	—	—
Not done - haven’t found a reliable	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Oct. 28, 2011	—	—	—	—	—

Worksheet: Point & Triangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`K10`	K10	`0`	`=E10`
`L10`	0	`0`	`=F10`
`K11`	K11	`8.3`	`=E11`
`L11`	8.3	`0`	`=F11`
`K12`	K12	`3.9`	`=E12`
`L12`	3.9	`4.3`	`=F12`
`C13`	C13	`A`	`=C10`
`E13`	E13	`0`	`=E10`

Cell Excerpt

Point - Projection onto Three Sides	—	—	—	—	—
Triangle ABC is formed	—	—	—	—	—
Point P is given	—	—	—	—	—
This sheet shows the coordinates of	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Nov. 4, 2011	—	—	—	—	—

Worksheet: barycentric (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F12`	0.5	`0.577350269189626`	`=SQRT(3)/3`
`D13`	D13	`A`	`=D10`
`E13`	A	`0`	`=E10`
`F13`	0	`0`	`=F10`
`D14`	D14	`B`	`=D11`
`E14`	B	`1`	`=E11`
`F14`	1	`0`	`=F11`
`D19`	D19	`a`	`=D16`

Cell Excerpt

Barycentric Coordinates	—	—	—	—	—
Analysis: Given a point, find the ba	—	—	—	—	—
Synthesis, given the barycentric coo	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Nov. 10, 2011 (Rachel’s birthday!)	—	—	—	—	—
Sept. 7, 2012	—	—	—	—	—

Worksheet: barycentric w CaR (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G32`	G32	`0.240422721268164`	`=G27/[.G$25]`
`G33`	G33	`0.383091149273448`	`=G28/[.G$25]`
`G34`	G34	`0.376486129458388`	`=G29/[.G$25]`
`G35`	G35	`1`	`=SUM(G34:G36)`

Cell Excerpt

Barycentric Coordinates

This is the spreadsheet calculationa

Create a reference triangle

Create a test point P

C.a.R. calculates subtriangle areas

Out of this should emerge the baryce

Worksheet: Trilinear (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`H11`	5	`12`	`=SUM(E11:G11)`
`I11`	12	`6`	`=H11/2`
`J11`	6	`60`	`=PRODUCT(E11:G11)`
`E12`	E12	`9`	`=E11^2`
`F12`	9	`16`	`=F11^2`
`G12`	16	`25`	`=G11^2`
`H12`	25	`50`	`=SUM(E12:G12)`
`I12`	50	`25`	`=H12/2`

Cell Excerpt

Trilinear Coordinates	—	—	—	—	—
Synthesis: Given the trilinear coord	—	—	—	—	—
Analysis: Given a point, find the tr	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Nov. 10, 2011	—	—	—	—	—
(Rachel’s birthday!)	—	—	—	—	—

Worksheet: trace (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`G16`	s	`2.3155`	`=SUM(G13:G15)/2`
`G17`	K	`1.0`	`=SQRT(G16(G16-G13)(G16-G14)*(G16-G15))`
`G18`	R	`0.9`	`=PRODUCT(G13:G15)/(4*G17)`
`H18`	0.9	`0.9`	`=G18`
`H21`	1	`1`	`=G21`
`H22`	1	`1`	`=G22`
`H23`	1	`1`	`=G23`
`G24`	atA	`#VALUE!`	`=[.F$70]*G21`

Cell Excerpt

Trace of a Point wrt a Triangle	—	—	—	—	—
The trace of a point P is the inters	—	—	—	—	—
The first line goes through P and on	—	—	—	—	—
The second line is the side of the t	—	—	—	—	—
Denote this point as TrA.	—	—	—	—	—
There are three trace points, or tra	—	—	—	—	—

Worksheet: Power (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`L5`	20.6	`0.811`	`=K9/25.4`
`L6`	32	`1.260`	`=K10/25.4`
`K7`	D	`63.5`	`=25.4*L11`
`K8`	R	`30.096773392205`	`=SQRT(K14^2-K9^2)`
`L8`	30.096773392205	`1.185`	`=K12/25.4`
`K10`	d1min	`36.4715748031496`	`=0.5*(K11-((K9^2-K10^2)/K11))`
`L10`	36.4715748031496	`1.436`	`=K14/25.4`
`K11`	d2min	`36.4715748031496`	`=0.5*(K11-((K9^2-K10^2)/K11))`

Cell Excerpt

Power of a Circle	—	—	—	—	—
Center of Similitudes and Radical Ax	—	—	—	—	—
zir file name:	—	—	—	—	—
—	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Nov. 11, 2011	—	—	Created	—	—

Worksheet: Intersecting Circles (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`L10`	20.6	`0.811`	`=K11/25.4`
`L11`	32	`1.260`	`=K12/25.4`
`K12`	D	`63.5`	`=25.4*L13`
`K13`	R	`30.096773392205`	`=SQRT(K16^2-K11^2)`
`L13`	30.096773392205	`1.185`	`=K14/25.4`
`K15`	d1min	`36.4715748031496`	`=0.5*(K13-((K11^2-K12^2)/K13))`
`L15`	36.4715748031496	`1.436`	`=K16/25.4`
`K16`	d2min	`36.4715748031496`	`=0.5*(K13-((K11^2-K12^2)/K13))`

Cell Excerpt

Intersecting Circles	—	—	—	—	—
Area of intersection. Length of chor	—	—	—	—	—
Radical axis location. Centers of si	—	—	—	—	—
Uses new format where K, L, M and N	—	—	—	—	—
zir file name:	—	—	—	—	—
Area of Overlapping Circles.zir	—	—	—	—	(weak)

Worksheet: Isotomic Conj (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`I12`	A	`0`	`=I9`
`J12`	0	`0`	`=J9`
`J13`	0.7	`0.7`	`=I13`
`J14`	0.7	`0.7`	`=I14`
`J15`	0.15	`0.15`	`=I15`
`I16`	D	`0.86`	`=I10+I13*(I11-I10)`
`J16`	0.86	`0.56`	`=J10+J13*(J11-J10)`
`I17`	E	`0.24`	`=I11+I14*(I12-I11)`

Cell Excerpt

Isotomic Conjugate	—	—	—	—	—
Points P and Q are isotomic conjugat	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Begun Sept. 23, 2012	—	—	—	—	—
Last modified Sept. 123, 2012	—	—	—	—	—
http://2000clicks.com/mathhelp/Geome	—	—	—	—	—

Worksheet: Harmonic Conj (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F15`	D	`-1`	`=1/((2/(F13-F12))-(1/F14))`
`F18`	k	`0.333333333333333`	`=(F14-F12)/(F13-F14)`
`G18`	0.333333333333333	`=`	`="="`
`F19`	k2	`0.25`	`=(F14-F12)/(F13-F12)`
`G19`	0.25	`=`	`="="`
`F20`	k3	`-0.5`	`=(F15-F12)/(F13-F12)`
`G20`	-0.5	`=`	`="="`
`F21`	k4	`0`	`=(F14-F38)/(F13-F12)`

Cell Excerpt

Harmonic Conjugate Pairs	—	—	—	—	—
Point pairs (A,B) and (C,D) are harm	—	—	—	—	—
\|AC\|/\|BC\| = \|AD\|/\|BD\|	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Begun Sept. 7, 2012	—	—	—	—	—
Last modified Sept. 12, 2012	—	—	—	—	—

Worksheet: HarmConj (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`H17`	H17	`5`	`=(H19+H18)/2`
`H18`	H18	`8`	`=G32+G21`
`H19`	H19	`8`	`=H24`
`I19`	8	`5`	`=SQRT([.$G$15]^2-[.$G$20]^2)`
`G25`	G25	`10`	`=H19-H18`
`G26`	G26	`5`	`=H23`
`G27`	G27	`8`	`=H18+G32+G21`
`L44`	L44	`0.2`	`=0.1+L49`

Cell Excerpt

Harmonic Conjugates	—	—	—	—	—
Applications: elliptical orbits	—	—	—	—	—
—	—	—	—	—	—
—	—	—	—	—	—
—	—	Radius of original circle	—	—	—
—	—	—	—	—	R

Worksheet: HC XYZ (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J8`	J8	`20`	`=(H10+H8)/2`
`K8`	20	`0.2`	`=H10/J8-1`
`L8`	0.2	`1.5`	`=(1+K8)/(1-K8)`
`H10`	X	`24`	`=H8*((H9+H8)/(H9-H8))`
`J11`	J11	`20`	`=(H13+H11)/2`
`K11`	20	`0.1`	`=H11/J11-1`
`L11`	0.1	`1.22222222222222`	`=(1+K11)/(1-K11)`
`H13`	Y	`18`	`=(SQRT((H11+H12)^2+4H11H12)-(H11+H12))/2`

Cell Excerpt

Harmonic Conjugates - Distances X, Y	—	—	—	—	—
X = \|AC\|, Y = \|CB\|, Z = \|BD\|	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Nov. 14, 2013	—	—	Created	—	—
Jun. 6, 2014	—	—	Updated	—	—
—	—	—	—	—	—

Worksheet: HarmMean (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J27`	4	`0.75`	`=H40/I40`
`J28`	7	`0.714285714285714`	`=H41/I41`
`F29`	H	`2.66666666666667`	`=2/((1/F40)+(1/F41))`
`H29`	H	`30`	`=2H40H41`
`I29`	30	`41`	`=I40H41+H40I41`
`J29`	41	`0.731707317073171`	`=H42/I42`
`J32`	3	`0.666666666666667`	`=H46/I46`
`J33`	7	`0.714285714285714`	`=H47/I47`

Cell Excerpt

Harmonic Mean	—
The harmonic mean of two numbers is	—
How it’s related to the harmonic con	—
Let x be the distance between A and	—
Then the diameter of the circle AB i	—
In general, the harmonic mean of n n	—

Worksheet: Orthogonal Circles (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J8`	r2 =	`0.5`	`=SQRT(J7^2-[.$G$8]^2)`
`I9`	I9	`1.1`	`=J7`
`J9`	1.1	`1.1`	`=J7`
`L9`	L9	`1.2`	`=M7`
`I10`	I10	`0.5`	`=J8`
`J10`	0.5	`0.5`	`=J8`
`L10`	L10	`0.7`	`=M8`
`E12`	0	`0`	`=2PI()(D12/36)`

Cell Excerpt

Orthogonal Circles	—	—	—	—	—
Given a unit circle centered at the	—	—	—	—	—
Find and plot circles that are ortho	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Begun August 1, 2012	—	—	—	—	—
Updated August 1, 2012	—	—	—	—	Unit Radius Circle

Worksheet: Orthogonal 2 (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J8`	r2 =	`0.5`	`=SQRT(J7^2-[.$G$8]^2)`
`I9`	I9	`1.1`	`=J7`
`J9`	1.1	`1.1`	`=J7`
`L9`	L9	`1.2`	`=M7`
`I10`	I10	`0.5`	`=J8`
`J10`	0.5	`0.5`	`=J8`
`L10`	L10	`0.7`	`=M8`
`E12`	E12	`0`	`=2PI()(D12/36)`

Cell Excerpt

Orthogonal Circles	—	—	—	—	—
Draw orthogonal circles of the form	—	—	—	—	—
Requires knowledge of radical axes	—	—	—	—	—
Fred Seelig	—	—	—	—	—
Begun August 2, 2012	—	—	—	—	—
Updated August 2, 2012	—	—	—	—	Unit Radius Circle

Worksheet: Isosceles (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`I19`	E	`0.3`	`=2*SIN(RADIANS(I18/2))`
`I20`	a	`0.08`	`=I19/(2*I17)`
`I21`	s	`2.2`	`=I17*(1+I20)`
`I22`	x	`0.2`	`=I20*I17`
`I23`	y	`0.2`	`=I22`
`I24`	z	`1.8`	`=I17*(1-I20)`
`I25`	K	`0.31`	`=SQRT(PRODUCT(I21:I24))`
`J25`	0.31	`0.31`	`=I20(I17^2)SQRT((1+I20)*(1-I20))`

Cell Excerpt

Isosceles Triangles

∆ is isosceles with two sides of len

For an N-sided regular polygon, E is

R are the pie cut lengths equal to t

Triangle parameter a = E/(2R)

s = R·(1 + a)

Worksheet: Golden Triangle (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F11`	F11	`0.09549`	`=F13^2`
`F12`	F12	`0.19098`	`=1-[.$F$36]/2`
`F13`	F13	`0.30902`	`=1/(2*[.F$36])`
`F14`	F14	`0.32492`	`=TAN(RADIANS(18))`
`F15`	\|IM\|	`0.36327`	`=F35/F42`
`F16`	\|GM\|	`0.51295`	`=F35/3`
`F17`	F17	`0.52786`	`=(F36-0.5)/(F36+0.5)`
`F18`	F18	`0.58779`	`=F31/2`

Cell Excerpt

The Golden Triangle	—	—	—	—	—
Isosceles triangle with equal sides	—	—	—	—	—
Unity base length, vertex angle 36°,	—	—	—	—	—
Fred W. Seelig	—	—	—	—	—
Mar. 22, 2015	—	—	Created	—	—
Mar. 31, 2015	—	—	Last Modified	—	—

Worksheet: Equil w Known Radials (Triangles and Circles.ods)

Cell Excerpt

Equilateral Triangle with Known Radi	—	—	—
An equilateral triangle ∆ABC with ed	—	—	—
contains point P somewhere in its in	—	—	—
PA, PB, and PC to its vertices	—	—	—
Find edge length E	—	—	—
Example: An equilateral triangle wit	—	—	—

Worksheet: Oai Equil ∆ (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`I27`	a	`7`	`=I28`
`I28`	b	`5`	`=I29`
`I29`	c	`6`	`=I30`
`I30`	a2	`49`	`=I28^2`
`I31`	b2	`25`	`=I29^2`
`I32`	c2	`36`	`=I30^2`
`I33`	a2	`49`	`=I34`
`I34`	b2	`25`	`=I35`

Cell Excerpt

The Oai Equilateral Triangle	—	—	—
Inside every acute triangle is an eq	—	—	—
whose vertices lay along the incente	—	—	—
Since the distances are different de	—	—	—
in order to make the equilateral tri	—	—	—
it is a matter of some art to find t	—	—	—

Worksheet: 20-40-120 (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J9`	20	`0.349065850398866`	`=RADIANS(I10)`
`J10`	40	`0.698131700797732`	`=RADIANS(I11)`
`I11`	g	`120`	`=180-(I10+I11)`
`J11`	120	`2.0943951023932`	`=RADIANS(I12)`
`I12`	a	`20`	`=I10`
`J12`	20	`0.349065850398866`	`=J10`
`I13`	b	`40`	`=I11`
`J13`	40	`0.698131700797732`	`=J11`

Cell Excerpt

The 20-40-120 Triangle	—	—	—	—	—
K = 1:	—	—	—	—	—
r = 1: (x,y,x) = (cot(a), …), (a,b,c	—	—	—	—	—
R = 1: extended law of sines: (a,b,c	—	—	—	—	—
Fred Seelig	—	—	—	—	—
7/16/2013	—	—	—	—	—

Worksheet: Euler 2x 3x angles (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`E17`	30°-60°-90° with hypotenuse = 1	`RIGHT`	`=IF(OR(L20>90;M20>90;N20>90);"OBTUSE";IF(AND(L20<90;M20<90;N20<90);"ACUTE";"RIGH`
`I18`	I18	`0.5`	`=SIN(RADIANS(L20))`
`J18`	0.5	`0.866025403784439`	`=SIN(RADIANS(M20))`
`I20`	I20	`1.73205080756888`	`=SQRT(3)`
`J20`	1.73205080756888	`1.73205080756888`	`=I22`
`K20`	1.73205080756888	`1.73205080756888`	`=J22`
`L20`	1.73205080756888	`60`	`=DEGREES(ACOS((J22^2+K22^2-I22^2)/(2J22K22)))`
`C23`	C23	`3`	`=C21+1`

Cell Excerpt

Euler Double and Triple Angle Triang	—	—	—	—	—
b = 2a	—	—	—	—	—
b = 3a	—	—	—	—	—
Fred W. Seelig	—	—	—	—	—
Created Jan. 1, 2014	—	—	—	—	—
Last Edited Jan 1, 2014	—	—	—	—	—

Worksheet: Special (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`F7`	F7	`2.36602540378444`	`=(3+SQRT(3))/2`
`E11`	Equilateral triangle with unit side	`ACUTE`	`=IF(OR(I12>90;J12>90;K12>90);"OBTUSE";IF(AND(I12<90;J12<90;K12<90);"ACUTE";"RIGH`
`L11`	60°	`1 1/2`	`=SUM(F11:H11)/2`
`F12`	F12	`1`	`=F11`
`G12`	1	`1`	`=G11`
`H12`	1	`1`	`=H11`
`I12`	1	`60`	`=DEGREES(ACOS((G12^2+H12^2-F12^2)/(2G12H12)))`
`J12`	60	`60`	`=DEGREES(ACOS((H12^2+F12^2-G12^2)/(2H12F12)))`

Cell Excerpt

Special Triangles	—	—	—	—	—
Fred W. Seelig	—	—	—	—	—
Last Edited: August 9, 2014	—	—	—	—	—
—	Enter value	—	—	—	—
—	problem	—	—	—	—
—	—	—	—	—	2.36602540378444

Worksheet: Hyperbolas of ∆ (Triangles and Circles.ods)

Cell Excerpt

Hyperbolas of the Reference Triangle	—	—	—	—	—
http://mathworld.wolfram.com/Kiepert	—	—	—	—	—
http://mathworld.wolfram.com/Feuerba	—	—	—	—	—
http://mathworld.wolfram.com/Jerabek	—	—	—	—	—
—	Kiepert	—	—	—	Feuerbach
—	A	—	—	—	A

Worksheet: Grk Sym (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`B1`	0	—	`=CHAR(A1)`
`D1`	32	—	`=CHAR(C1)`
`F1`	64	`@`	`=CHAR(E1)`
`H1`	96	`'`	`=CHAR(G1)`
`J1`	128	`�`	`=CHAR(I1)`
`L1`	160	`�`	`=CHAR(K1)`
`A2`	A2	`1`	`=A1+1`
`B2`	1	—	`=CHAR(A2)`

Cell Excerpt

0	—	32	—	64	@
1	—	33	!	65	A
2	—	34	“	66	B
3	—	35	#	67	C
4	—	36	$	68	D
5	—	37	%	69	E

Worksheet: E2 ASCII (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`C4`	1	`3`	`=[.C$2]+SQRT([.$B$1]^2-(B4-[.B$2])^2)`
`B5`	B5	`1.2`	`=B4+0.2`
`C5`	1.2	`3.87177978870813`	`=[.C$2]+SQRT([.$B$1]^2-(B5-[.B$2])^2)`
`B6`	B6	`1.4`	`=B5+0.2`
`C6`	1.4	`4.2`	`=[.C$2]+SQRT([.$B$1]^2-(B6-[.B$2])^2)`
`B7`	B7	`1.6`	`=B6+0.2`
`C7`	1.6	`4.42828568570857`	`=[.C$2]+SQRT([.$B$1]^2-(B7-[.B$2])^2)`
`B8`	B8	`1.8`	`=B7+0.2`

Cell Excerpt

Radius	2	—
Center	3	3
—	x	y
—	1	3
—	1.2	3.87177978870813
—	1.4	4.2

Worksheet: circle templ (Triangles and Circles.ods)

Key Formulas & Parameters

Cell	Label / Context	Value	OpenFormula
`J4`	J4	`1/16`	`=[.$C6]&" "&G6&"/"&[.G$5]`
`K4`	1/16	`0.0625`	`=C6/[.C$5]+D6/[.D$5]+E6/[.E$5]+F6/[.F$5]+G6/[.G$5]+H6/[.H$5]+I6/[.I$5]`
`B5`	B5	`2`	`=B6+1`
`J5`	5	`5/64`	`=[.$C7]&" "&I7&"/"&[.I$5]`
`K5`	5/64	`0.0781`	`=C7/[.C$5]+D7/[.D$5]+E7/[.E$5]+F7/[.F$5]+G7/[.G$5]+H7/[.H$5]+I7/[.I$5]`
`B6`	B6	`3`	`=B7+1`
`J6`	J6	`3/32`	`=[.$C8]&" "&H8&"/"&[.H$5]`
`K6`	3/32	`0.0938`	`=C8/[.C$5]+D8/[.D$5]+E8/[.E$5]+F8/[.F$5]+G8/[.G$5]+H8/[.H$5]+I8/[.I$5]`

Cell Excerpt

Circle Templates	—	—	—	—	—
—	—	1	2	4	8
—	1	—	—	—	—
—	2	—	—	—	—
—	3	—	—	—	—
—	4	—	—	—	—

0	—	32	—	64	@
1	—	33	!	65	A
2	—	34	“	66	B
3	—	35	#	67	C
4	—	36	$	68	D
5	—	37	%	69	E

0	—	32	—	64	@
1	—	33	!	65	A
2	—	34	“	66	B
3	—	35	#	67	C
4	—	36	$	68	D
5	—	37	%	69	E

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Master Worksheet Index (Triangles and Circles.ods)¶

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Master Worksheet Index (`Triangles and Circles.ods`)¶

0	—	32	—	64	@
1	—	33	!	65	A
2	—	34	“	66	B
3	—	35	#	67	C
4	—	36	$	68	D
5	—	37	%	69	E