X3758–X3861

Central Triangle Combos Xcom(T) II: Parameterized T(f, g) & Euler Triangles

Combo centers Xk-com[T(f, g)] and Xk-com[Inverse(nT(f, g))] for parameterized central triangles (X3758–X3810), followed by combo centers on the 1st through 5th Euler triangles (X3811–X3861).
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) ( A R_b ): Euclid I.22 Guide Circle (A) ( B R_a ): Euclid I.22 Guide Circle (B) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) circumcircle: Circumcircle of △ABC incircle: Incircle of △ABC nine_point_circle: Nine-Point Circle (Euler / Feuerbach Circle) { 2, 3 }: Euler Line A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) R_b = Radius Point R_b (0.8571, 0.0000) R_a = Radius Point R_a (0.2857, 0.0000) C = Vertex C (0.6122, 0.5999) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_1 = INCENTER (0.5714, 0.2333) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_2 = CENTROID (0.5374, 0.2000) X_3758 = com[T(bc, b2)] (0.5669, 0.2348) X_3759 = com[T(-bc, b2)] (0.6959, 0.3610) X_3760 = com[T(-bc, c2)] (0.3482, 0.0495) X_3761 = com[T(bc, c2)] (0.4942, 0.1697) X_3769 = com[T(-bc, b2)] (0.8099, 0.4837) X_3781 = com[INVERSE(nT(bc, c2))] (0.4626, 0.0816) X_3799 = com[INVERSE(nT(bc, c2))] (0.0667, 0.4044) X_3800 = com[INVERSE(nT(a2, a2 + b2))] (1.0700, -0.0464) X_3802 = com[INVERSE(nT(bc, c2))] (0.3415, 0.3331) X_3807 = com[INVERSE(nT(bc, b2))] (0.0074, 0.3339) X_3808 = com[INVERSE(nT(-bc, c2))] (0.1622, -0.2513) X_3810 = com[INVERSE(nT(-bc, c2))] (1.1140, 0.0906) X_3814 = com(EULER TRIANGLE) (0.1395, -0.5166) X_3823 = com(4th EULER TRIANGLE) (0.2585, -0.1278) X_3827 = com(2nd EULER TRIANGLE) (0.4466, -0.3798) X_3830 = com(EULER TRIANGLE) (0.6497, 0.4936) A B C X3 X1 X5 X2 X3758 X3759 X3760 X3761 X3769 X3781 X3799 X3800 X3802 X3807 X3808 X3810 X3814 X3823 X3827 X3830 (a, b, c) = (5, 6, 7) | AB = 1

X3758–X3861 (Central Triangle Combos Xcom(T) II: Parameterized T(f, g) & Euler Triangles) — 16 representative centers (from 104 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Central Triangles and More Combos

Many triangle centers can be defined (as just above) by the form Xcom(nT), where T denotes a central triangle. In order to present such centers, it is helpful to introduce the notation T(f(a,b,c), g(b,c,a)) for central triangles. Following TCCT, pages 53-54, suppose that each of f(a,b,c) and g(a,b,c) is a center-function or the zero function, and that one of these three conditions holds:

the degree of homogeneity of g equals that of f;
f is the zero function and g is not the zero function;
g is the zero function and f is not the zero function.

There are two cases to be considered: If \(g(a,b,c)=g(a\),c,b), then the central triangle T(f(a,b,c), g(b,c,a)) is defined by the following 3x3 matrix (whose rows give homogeneous coordinates for the A-, B-, C- vertices, respectively):

f(a,b,c) g(b,c,a) g(c,a,b)
g(a,b,c) f(b,c,a) g(c,a,b)
g(a,b,c) g(b,c,a) f(c,a,b)

If g(a,b,c) is not equal to g(a,c,b), then the central triangle T(f(a,b,c),g(b,c,a)) is defined by the following 3x3 matrix:

f(a,b,c) g(b,c,a) g(c,b,a)
g(a,c,b) f(b,c,a) g(c,a,b)
g(a,b,c) g(b,a,c) f(c,a,b)

If the homogenous coordinates are chosen to be barycentric, as they are for present purposes, then the new notation for central triangles is typified by these examples:

Medial triangle = T(1, 0)
Anticomplementary triangle = T(-1, 1)
Incentral triangle = T(a, 0)
Excentral triangle = T(-a, b)
Euler triangle = T(2(\(b^{4}\) + \(c^{4}\) - \(b^{2}\)\(c^{2}\) - \(c^{2}\)\(a^{2}\) - \(a^{2}\)\(b^{2})\), (\(b^{2}\) + \(c^{2}\) - \(a^{2})\)(\(a^{2}\) + \(b^{2}\) - \(c^{2})\))
2nd Euler triangle = \(T(2a^{2}(b^{4} + c^{4} - 2b^{2}c^{2} - c^{2}a^{2} - a^{2}b^{2})\), (\(a^{2}\) - \(b^{2}\) + \(c^{2})\)(\(a^{4}\) + \(b^{4}\) + \(c^{4}\) - \(2a^{2}\)\(c^{2}\) - \(2b^{2}\)\(c^{2})\))
3rd Euler triangle = T((b - c)2, \(c^{2}\) - \(a^{2}\) - bc)
4th Euler triangle = T((b + c)2, \(c^{2}\) - \(a^{2}\) + bc)
5th Euler triangle = T(2(\(b^{2}\) + \(c^{2})\), \(b^{2}\) + \(c^{2}\) - \(a^{2}\) )

The vertices of the five Euler triangles lie on the nine-point circle. (See C. Kimberling, “Twenty-one points on the nine-point circle,” Mathematical Gazette 92 (2008) 29-38.) Randy Hutson observes (January 8, 2015) that the 2nd Euler triangle is the complement of the circumorthic triangle, the 2nd Euler triangle is the reflection of the orthic triangle in X5; the 3rd Euler triangle is the complement of the 1st circumperp triangle, the 4th Euler triangle is the complement of the 2nd circumperp triangle; the 4th Euler triangle the triangle whose sidelines are the radical axes of each of the Odehnal tritangent circles (defined at X6176) and the corresponding excircle; the 5th Euler triangle is the complement of the circummedial triangle, and the vertices of the 5th Euler triangle are the intersections, other than midpoints of the sides of ABC, of the nine-point circle and the medians.

The notation T(f(a,b,c),g(b,c,a)) can be used to define several more triangles; in each case, two of the perspectivities can be used to construct the triangle.

T(bc, \(b^{2})\) is perspective to ABC with perspector X6; other such pairs: incentral, X192, excentral, X1045; tangential, X6; anticomplementary, X192

T(-bc, \(b^{2})\) is perspective to ABC with perspector X6; incentral, X2; tangential, X6; T(bc,\(b^{2})\), X6

T(bc, \(c^{2})\) is perspective to ABC with perspector X76; anticomplementary, X8; 4th Brocard, X76, Fuhrmann, X8; cevian <cevian>`(:ref:`X350),X1

T(-bc, \(c^{2})\) is perspective to ABC with perspector X76; cevian(X321),X75; anticevian(X8), X2

\(T(a^{2}\), \(a^{2}\) - \(b^{2})\) is perspective to the medial triangle with perspector X3; tangential , X3; symmedial, X2; cevian(X287), X98

\(T(a^{2}\), \(a^{2}\) - \(c^{2})\) is perspective to the tangential triangle with perspector X25; orthic , X25; medial, X6; 1st Lemoine, X1383; circummedial, X251

\(T(a^{2}\), \(a^{2}\) + \(b^{2})\) is perspective to ABC with perspector X83; medial, X6; 1st Neuberg, X182

\(T(a^{2}\), \(a^{2}\) + \(c^{2})\) is perspective to ABC with perspector X141; medial, X3; 1st Neuberg, X182; \(T(a^{2}\),\(a^{2}\) - \(c^{2})\), X2; cevian(X69), X6; cevian(X76), X2; cevian(X3313),X39

The triangles IT1 and IT2 are triply perspective to ABC (as are T1 and T3). Order-label IT1 as A'B'C'. The three perspectivities are then given by

AA'∩BB'∩CC' = X3862
AB'∩BC'∩CA' =\(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = ab(a^{2} - bc)(c^{2} - ab)(b^{2} + bc + c^{2})\)
AC'∩BA'∩CB' =\(f(a,c,b) : f(b,a,c) : f(c,b,a)\)

Peter Moses noted (12/23/2011) that T(bc, \(b^{2})\) is triply perspective to ABC. With T(bc, \(b^{2})\) order-labeled as A'B'C', the three perspectivities with perspectors are as follows:

AA'∩BB'∩CC' = X6; AB'∩BC'∩CA' = P(6); AC'∩BA'∩CB' = U(6). (The notation P(k) and U(k) refers to a bicentric pair of points; see Tables at the top of ETC.)
Likewise, T(bc, \(c^{2})\) is triply perspective to ABC, with perspectors: X76, P(10), and U(10).

(This section was added to ETC on 12/23/2011.)

X_3767 — X_3053com[T(a², a² - c²)]

X3767com[T(a2, a2 - c2)]

X3767 is the center of the inellipse that is the barycentric square of the orthic axis. The Brianchon point (perspector) of the inellipse is X393. (Randy Hutson, October 15, 2018)
X_3779 — X_3056com[T(-bc, b²)]

X3779com[T(-bc, b2)]

X3779 = homothetic center of extangents triangle and reflection of intangents triangle in X6
X_3822 — X_2099com(3rd EULER TRIANGLE)

X3822com(3rd EULER TRIANGLE)

Suppose that ABC is an acute triangle. Let LA be the circle that is externally tangent to the nine-point circle and to the sidelines AB and AC. Define LB and LC cyclically. The three circles are here named the Odehnal tritangent circles. Their radical center is X3822. The center of the Apollonian circle of LA, LB, LC is X6167. (Boris Odehnal, “A Triad of Tritangent Circles, Journal for Geometry and Graphics 18 (2014) no. 1, 61-71)
X_3830 — X_3543com(EULER TRIANGLE)

X3830com(EULER TRIANGLE)

X3830 = QA-P24 (AntiComplement of QA-P1 wrt the Morley Triangle) of quadrangle ABCX2 (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/52-qa-p24.html)