X176

EQUAL DETOUR POINT

A point X is defined as a point of equal detour of triangle ABC if |XB| + |XC| - |BC| = |XC| + |XA| - |CA| = |XA| + |XB| - |AB|. Veldkamp established that X = X176 for some triangles ABC, but the conditions he gives are not correct. Hajja and Yff proved that the condition tan(A/2) + tan(B/2) + tan(C/2) < 2 is necessary and sufficient for the existence of exactly two points of equal detour and that the condition tan(A/2) + tan(B/2) + tan(C/2) = 2 is necessary and sufficient for the existence of exactly one point of equal detour. Yff found that X176 is also is the center of the inner Soddy circle. See also X175 and the 1st and 2nd Eppstein points, X481, X482.
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 1, 7 }: Soddy Line { 489, 664 }: Central Line L(489, 664) { 8, 1271 }: Central Line L(8, 1271) { 226, 1132 }: Central Line L(226, 1132) { 174, 1274 }: Central Line L(174, 1274) { 2, 1659 }: Central Line L(2, 1659) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1 = INCENTER (0.5714, 0.2333) X_7 = GERGONNE POINT (0.5903, 0.2769) X_489 = CEVAPOINT OF X(20) AND X(487) (0.4404, 0.3605) X_664 = TRILINEAR POLE OF LINE X(2)X(7) (0.2391, 0.5142) X_8 = NAGEL POINT (0.4694, 0.1333) X_1271 = ISOTOMIC CONJUGATE OF X(1132) (-0.0936, -0.4782) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_1132 = ARCTAN(-2) KIEPERT POINT (0.8864, -0.3774) X_174 = YFF CENTER OF CONGRUENCE (0.5813, 0.2549) X_1274 = ISOTOMIC CONJUGATE OF X(1143) (0.4773, 0.1423) X_2 = CENTROID (0.5374, 0.2000) X_1659 = YIU-PAASCHE POINT (0.5749, 0.2470) X_176 = EQUAL DETOUR POINT (0.5803, 0.2537) A B C X1 X7 X489 X664 X8 X1271 X226 X1132 X174 X1274 X2 X1659 X176 (a, b, c) = (5, 6, 7) | AB = 1

X176 (EQUAL DETOUR POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1 + \sec A/2 \cos B/2 \cos C/2 : :\)
Barycentrics: \((\sin A)(1 + \sec A/2 \cos B/2 \cos C/2)\)
Search Value (6, 9, 13): \(1.190492903652\)
▸ Alternate Coordinates3 forms

  • Barycentrics: \(2a + (a + b + c) \tan(A/2) : :\)

  • Barycentrics: \(a + r_{a} : b + r_{b} : c + r_{c}\), where \(r_{a}\), \(r_{b}\), \(r_{c}\) are the exradii

  • Barycentrics: \((a + b - c) \cdot (a - b + c) \cdot (a (a - b - c) - S) : :\)

▸ Linear Combinations1 combination

  • X176 = \(2s\) X1 + \((r + 4R)\) X7 = \(3\) X2 - \(4\) X31535

X176 lies on K032, K199, K200, K1175, Q074, Q092, Q104.

▸ Incident Central Lines61 lines

{1, 7}, {2, 1659}, {4, 10904}, {8, 1271}, {40, 34494}, {65, 6404}, {69, 10907}, {105, 30386}, {144, 30556}, {174, 1274}, {226, 1132}, {241, 38487}, {280, 40699}, {329, 3083}, {388, 10910}, {489, 664}, {517, 39794}, {651, 1124}, {1086, 44635}, {1123, 10252}, {1463, 7362}, {1587, 8953}, {1588, 8978}, {2082, 6204}, {2550, 45714}, {3062, 10972}, {3084, 9776}, {3177, 31408}, {3296, 34215}, {3297, 6180}, {3298, 5228}, {3622, 8243}, {4000, 7969}, {4644, 7968}, {5045, 39795}, {5222, 18991}, {5226, 5405}, {5261, 9906}, {5393, 5435}, {5932, 31529}, {5933, 31531}, {6283, 20358}, {6360, 37881}, {6405, 49537}, {7586, 30324}, {8973, 23259}, {9778, 10134}, {10405, 14121}, {10580, 16662}, {11294, 17086}, {13386, 20211}, {13388, 21454}, {13437, 16440}, {15913, 31537}, {16213, 32080}, {17365, 44636}, {17950, 43133}, {21453, 30336}, {30346, 31589}, {30412, 31548}, {31545, 32200}

▸ Functional Relations20 relations

Notes

The points X175 and X176 are discussed in an 1890 article by Emile Lemoine, accessible at Gallica. The article begins on page 111, and the two points are considered beginning on page 128.

The following construction was found by Elkies: call two circles within ABC companion circles if they are the incircles of two triangles formed by dividing ABC into two smaller triangles by passing a line through one of the vertices and some point on the opposite side; chain of circles O(1), O(2), … such that O(n),O(n+1) are companion incircles for every n consists of at most six distinct circles; there is a unique chain consisting of only three distinct circles; and for this chain, the three subdividing lines concur in X176.

G. R. Veldkamp, “The isoperimetric point and the point(s) of equal detour,” American Mathematical Monthly 92 (1985) 546-558.

Noam D. Elkies and Jiro Fukuta, Problem E 3236 and Solution, American Mathematical Monthly 97 (1990) 529-531 [proposed 1987].

Mowaffaq Hajja and Peter Yff, “The isoperimetric point and the point(s) of equal detour in a triangle,” Journal of Geometry 87 (2007) 76-82.

Let Ia, Ib, Ic be the centers of the Elkies companion incircles. Let A' be the trilinear product Ib*Ic, and define B' and C' cyclically. The lines AA', BB', CC' concur in X482. The lines IaA', IbB', IcC' concur in X176. (Randy Hutson, December 2, 2017)

Let Q be a point. Let A' be the incenter of triangle BCQ, and define B' and C' cyclically. If Q is an equal detour point, in the sense that |QB| + |QC| - |BC| = |QC| + |QA| - |CA| = |QA| + |QB| - |AB|, then Q is either the incenter or an excenter of triangle A'B'C'. (Do there exist other points with this property?) (Randy Hutson, 9/23/2011)’’

For the Moses-Gheorghe-Reznick conic, see X175.

In the plane of a triangle ABC, let

Oa = circle with diameter BC

C' = the point, other than B, where Oa meets AB

B' = the point, other than C, where Oa meets AC

Ub = B-mixtilinear incircle of BCC'

Uc = C-mixtilinear incircle of CBC'

Ab = touchpoint of Ub and BC; define Bc and Ca cyclically

Ac = touchpoint of Uc and BC; define Ba and Cb cyclically

The points Ab, Ac, Ba, Ba, Cb, Cb line on a conic, of which the center is X176. A barycentric equation for this conic follows:

2 (a-b-c) (a^2-b^2-c^2) (a^2 (b+c)-(b-c)^2 (b+c)+2 a S) x^2 - ((a+b-c) (a-b+c) (3 a^4-2 a^3 (b+c)-3 (b^2-c^2)^2+2 a (b+c) (b^2+c^2))+2 (a^4-3 (b^2-c^2)^2+2 a^2 (b^2+c^2)) S) y z + (cyclic) = 0.

See X175 and X176 (X(175)andX(176).png). Continuing, let

Vb = B-mixtilinear excircle of BCC'

Vc = C-mixtilinear excircle of CBC'

A'b = touchpoint of Vb and BC; define B'c and C'a cyclically

A'c = touchpoint of Vc and BC; define B'a and C'b cyclically

The points A'b, A'c, B'a, B'c, C'a, C'b line on a conic, of which the center is X175. A barycentric equation for this conic follows:

(a+b-c) (a-b+c) (2 (a-b-c) (2 a^3 b c+a^4 (b+c)-2 a b c (b+c)^2+(b-c)^2 (b+c)^3-2 a^2 (b^3+b^2 c+b c^2+c^3)+2 (a^3-2 b c (b+c)-a (b+c)^2) S) x^2 + (3 a^6-4 a^5 (b+c)-4 a (b-c)^2 (b+c)^3-3 (b-c)^2 (b+c)^4+a^2 (b+c)^2 (9 b^2+2 b c+9 c^2)-a^4 (9 b^2+14 b c+9 c^2)+8 a^3 (b^3+b^2 c+b c^2+c^3)-2 (a^4+6 a^3 (b+c)-4 a^2 (b+c)^2+3 (b^2-c^2)^2-2 a (3 b^3+7 b^2 c+7 b c^2+3 c^3)) S) y z) + (cyclic) = 0. (Angel Montesdeoca, September 9, 2022)