Unit IV: Books III & IV — Circles, Tangents & Regular Polygons

Cyclic Angles, Power of a Point, The Golden Triangle & The Pentagon (Pickering pp. 73–151)

[Folio III.20 & III.31 | p. 102]

Book III, Propositions XX & XXXI (Inscribed Angles & Thales’s Theorem, pp. 102, 115)

Book III, Prop. XX — Central & Inscribed Angles ( O A ) < B A C ) < B O C ) < O B A ) / A O / / O D / / A B / / A C / / O B / / O C / O = (0, 0) A = (0, 5) D = (0, -5) B = (-4, -3) C = (3, -4)
**Book III, Prop. XX (p. 102 · G.185).** The angle at the centre :byrne-angle:`blue ` is double the angle at the circumference :byrne-angle:`yellow `.

Prop. XX Demonstration (:just:`I.5`, :just:`I.32`). Drawing diameter + , isosceles radii make base angles equal to the corresponding part of , so each exterior central sector doubles its inscribed half;

∴ = 2 × . Q.E.D.

Book III, Prop. XXXI — Angle in a Semicircle (Thales) ( O B ) < B A O ) < O A C ) < O B A ) < A C O ) / B O / / O C / / O A / / B A / / A C / O = (0, 0) B = (-5, 0) C = (5, 0) A = (-3, 4)
**Book III, Prop. XXXI (p. 115 · G.196).** Thales's Theorem: the angle in a semicircle (:byrne-angle:`yellow ` + :byrne-angle:`black `) is a right angle (:byrne-op:`right-angle`).

Prop. XXXI Demonstration (:just:`I.5`, :just:`I.32`). Since radii = = , isosceles triangles give = and = . By I.32, ( + ) equals half of = . Q.E.D.

[Folio III.36 | p. 121]

Book III, Proposition XXXVI (Tangent-Secant Power of a Point, p. 121 · G.201)

Book III, Prop. XXXVI — Tangent-Secant Power of a Point ( O A ) < D T O ) / D T / / D C / / C O / / O A / / O T / O = (0, 0) A = (3, 0) C = (-3, 0) D = (-5, 0) T = (-9/5, 12/5)
**Book III, Prop. XXXVI (p. 121).** Square on tangent :byrne-line:`blue solid ` equals the rectangle contained by whole secant and external part :byrne-line:`red solid `.

Demonstration (:just:`III.18`, :just:`II.6`, :just:`I.47`). Because contact radius ⊥ tangent ( = ), applying II.6 to the bisected diameter ( + ) plus external yields:

(whole secant × ) + 2 = ( + )² = 2 + 2.

Canceling equal radius squares ( = ), ∴ whole secant × = 2. Q.E.D.

[Folio IV.4, IV.10 & IV.11 | p. 129]

Book IV, Propositions IV, X & XI (Incircle, Golden Triangle & Regular Pentagon, pp. 129–137)

Book IV, Prop. IV — Incircle of a Triangle < A B C > ( I E ) / A B / / A C / / B C / / B I / / C I / / I E / / I F / / I G / B = (0, 0) C = (14, 0) A = (5, 12) I = (6, 4) E = (6, 0) F = (30/13, 72/13) G = (46/5, 32/5)
**Book IV, Prop. IV (p. 129 · G.217).** Inscribing :byrne-circle:`black ` inside a given triangle (:byrne-line:`yellow solid `, :byrne-line:`red solid `, :byrne-line:`black solid `).
Book IV, Prop. X — The 36°–72°–72° Golden Triangle < A B D > ( A B ) ( P A ) < B A D ) < C D B ) < D B A ) / A C / / C B / / A D / / B D / / C D / A = (0, 0) B = (1, 0) C = (-1/2 + sqrt(5)/2, 0) D = (1/4 + sqrt(5)/4, sqrt(2)*sqrt(5 - sqrt(5))/4)
**Book IV, Prop. X (pp. 135–136 · G.223).** The $36^\circ\text{–}72^\circ\text{–}72^\circ$ Golden Triangle uniting ``II.11`` and ``III.37``.

Prop. X Synthesis (:just:`II.11`, :just:`IV.5`, :just:`III.37`, :just:`III.32`). Cut radius + in golden ratio ([II.11]), fit base chord = inside , and circumscribe . Since 2 equals whole radius × , is tangent to ([III.37]), whence = apex ([III.32]) and each base angle = 2 × . Q.E.F.

Book IV, Prop. XI — Regular Pentagon Inscribed in a Circle < A C D > < A B C D E > ( O A ) < C A D ) < E C A ) < D C E ) < B D C ) / A B / / B C / / C D / / D E / / E A / / C E / / D B / A = (0, 1) B = (-sqrt(2)*sqrt(sqrt(5) + 5)/4, -1/4 + sqrt(5)/4) C = (-sqrt(2)*sqrt(5 - sqrt(5))/4, -sqrt(5)/4 - 1/4) D = (sqrt(2)*sqrt(5 - sqrt(5))/4, -sqrt(5)/4 - 1/4) E = (sqrt(2)*sqrt(sqrt(5) + 5)/4, -1/4 + sqrt(5)/4)
**Book IV, Prop. XI (p. 137 · G.224).** Regular pentagon inscribed in :byrne-circle:`blue ` by bisecting the base angles of inscribed Golden Triangle :byrne-tri:`yellow filled >`.

Prop. XI Demonstration (:just:`IV.10`, :just:`IV.2`, :just:`I.9`, :just:`III.26`, :just:`III.29`).

Bisecting both $72^\circ$ base angles of by star diagonals produces five equal $36^\circ$ inscribed angles (, , , ), subtending five equal arcs and five equal pentagon chords: = = = = . Q.E.F.