X3663–X3757

Central Triangle Combos Xcom(T) I: Intouch, Extouch, Incentral & Hexyl Triangles

Combo centers Xk-com(T) evaluated on the intouch (X3663–X3677), extouch (X3678–X3719), incentral (X3720–X3728), excentral, tangential, 1st Brocard, and hexyl triangles (X3729–X3738), plus matrix-product combos n(incentral)*n(medial) and inverse-triangle combos (X3739–X3757).
< 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 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_3663 = com(INTOUCH TRIANGLE) (0.5625, 0.2127) X_3667 = com(INTOUCH TRIANGLE) (0.9481, -0.2192) X_3668 = com(INTOUCH TRIANGLE) (0.6162, 0.3370) X_3678 = com(EXTOUCH TRIANGLE) (0.4786, 0.1283) X_3680 = com(EXTOUCH TRIANGLE) (0.2679, 0.0583) X_3682 = com(EXTOUCH TRIANGLE) (0.3587, 0.0249) X_3693 = com(EXTOUCH TRIANGLE) (-0.4771, -0.6338) X_3699 = com(EXTOUCH TRIANGLE) (-0.0612, 0.2666) X_3708 = com(EXTOUCH TRIANGLE) (0.7157, 0.0317) X_3717 = com(EXTOUCH TRIANGLE) (-0.2517, -0.3555) X_3722 = com(INCENTRAL TRIANGLE) (0.4643, 0.2916) X_3724 = com(INCENTRAL TRIANGLE) (1.0944, 1.3190) X_3732 = com(EXCENTRAL TRIANGLE) (0.1895, 0.5356) X_3738 = com(HEXYL TRIANGLE) (0.0290, -0.0931) X_3747 = com[n(MEDIAL)*n(INCENTRAL)] (1.3019, 1.1933) X_3756 = com[INVERSE(n(HEXYL TRIANGLE))] (0.8367, 0.1666) A B C X3 X1 X3663 X3667 X3668 X3678 X3680 X3682 X3693 X3699 X3708 X3717 X3722 X3724 X3732 X3738 X3747 X3756 (a, b, c) = (5, 6, 7) | AB = 1

X3663–X3757 (Central Triangle Combos Xcom(T) I: Intouch, Extouch, Incentral & Hexyl Triangles) — 16 representative centers (from 95 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Group Preamble — Combos Using Central Triangles (Central Triangles and More Combos)

The discussion of combos near the beginning of ETC is continued here. Suppose that T is a central triangle, and let nT its normalization, so that the triangle nT is essentially a 3x3 matrix with row sums equal to 1, and the rows of nT are normalized barycentrics for the A-, B-, C- vertices of T.

Let X be a triangle center, given by barycentrics \(x : y : z\), not necessarily normalized. The point whose rows are the matrix product X*(nT) is then a triangle center, denoted by Xcom(T).

Among central triangles T are cevian and anticevian triangles and others described at MathWorld. A brief list follows, with A-vertices given in barycentrics (not normalized):

Intouch triangle = cevian triangle of X7
A-vertex = 0 : 1/(c + a - b) : 1/(a + b - c)
Extouch triangle = cevian triangle of X8
A-vertex = 0 : c + a - b : a + b - c
Incentral triangle = cevian triangle of X1
A-vertex =\(0 : b : c\)
Excentral triangle = anticevian triangle of X1
A-vertex = \(-a : b : c\)
Hexyl triangle
A-vertex =\(a(1 + a_{1} + b_{1} + c_{1}) : b(-1 + a_{1} + b_{1} - c_{1}) : c(1 + a_{1} - b_{1} + c_{1})\), where \(a_{1}\) = cos A, \(b_{1}\) = cos B, \(c_{1}\) = cos C
Tangential triangle = anticevian triangle of X6
A-vertex = \(-a^{2} : b^{2} : c^{2}\)
1st Brocard triangle
A-vertex =\(a^{2} : c^{2} : b^{2}\)

Group Preamble — More Combos (More Combos. Continuing the discussion of points Xcom(T))

Continuing the discussion of points Xcom(T), suppose that T is an arbitrary triangle, and let nT denote the normalization of T. Let NT denote the set of these triangles, as matrices, and let * denote matrix multiplication. Then NT is closed under *. Also, NT is closed under matrix inversion. Consequently, (NT, *) is a group, comparable to the group of stochastic matrices.

Suppose that T1 and T2 are triangles. In many cases, the product T1*T2 is well-defined (e.g., TCCT, page 175). However, n(T1*T2) may not be n(T1)*n(T2) if T1 and T2 are not normalized. Therefore, it is important, when dealing with products, to include the “n” if it is intended.

As a class of examples of triangles defined by matrix products, suppose that T1 is the cevian triangle of a triangle center \(f : g : h\) (barycentrics) and that T2 is the cevian triangle of a triangle center \(u : v : w\). The A-vertex of the triangle T3 = (\(nT_{1})\)*(\(nT_{2})\) is given by

\(u(gu + hu + gv +hw) : hv(u + w) : gw(u + v)\),

from which it can be shown that T3 is perspective to the triangle ABC, with perspector

P =\(ghu(v + w) : hfv(w + u) : fgw(u + v)\).

Also, T3 is perspective to T2, with perspector

Q = u(v + w)(gu + hu + gv + hw) : v(w + u)(hv + fv + hw + fu) : w(u + v)(fw + gw + fu + gv).

Thus, T3 can be constructed directly from the pairs {P, ABC} and {Q, T2}. Regarding the possibility that T3 is also perspective to T1, the concurrence determinant for this condition factors as F1F2F3, where

F1 = u + v + w,
F2 = fu(g + h) + gv(h + f) + hw(f + g),
F3 = (fghuvw)2[(gu)-2 - (hu)-2 + (hv)-2 - (fv)-2 + (fw)-2 - (gw)-2].

Consequently, the perspectivity holds if and only if one of the three factors is 0, and from this result, various geometric results can be obtained.

Example: The Incentral and Medial triangles result by taking \(f : g : h\) =\(a : b : c\) and \(u : v : w\) =\(1 : 1: 1\). The product n(Incentral)*n(Medial) has vertices

A' = b +\(c : c : b\) B' = c : c + a : a C' =\(b : a : a\) + b

and its inverse has A-vertex A″ = bc + ca + ab : -bc - ca + ab : -bc + ca - ab, from which B″ and C″ are obtained cyclically.

The product n(Medial)*n(Incentral) has A-vertex \(a(2a + b + c) : b(c + a) : c(a + b)\), and the inverse of this product has A-vertex (a + b + c)(b + c) : -c(a + c) : -b(a + b).

Having considered the group NT of normalized triangles, we turn next to 3-point combos based on product triangles and inverse triangles. Recall that such a combo, denoted by Xcom(T), is defined, as in the preamble to X3663, by the matrix product (x y z)*(nT), where nT is the normalization of T.

X_3671 — X_958com(INTOUCH TRIANGLE)

X3671com(INTOUCH TRIANGLE)

Let O* be a circle with center X2 and variable radius R*. Let LA be the radical axis of O* and the A-excircle, and define LB and LC cyclically. Let A'=LB∩LC, and define B' and C' cyclically. There is a unique value of R* for which triangle A'B'C' is perspective to ABC, and the perspector is X3671. (Randy Hutson, January 15, 2019)
X_3734 — X_6com(1st BROCARD TRIANGLE)

X3734com(1st BROCARD TRIANGLE)

Let A'B'C' be the 1st Brocard triangle. Let A″ be the trilinear pole, with respect to A'B'C', of line BC, and define B″ and C″ cyclically. The lines A'A″, B'B″, C'C″ concur in X3734.
X_3751 — X_1721com[INVERSE(n(HEXYL TRIANGLE))]

X3751com[INVERSE(n(HEXYL TRIANGLE))]

Let A'B'C' denote the inverse of the normalized hexyl triangle. Then A', expressed in homogeneous barycentrics (i.e., not normalized), is given by