X1142

1st MALFATTI-RABINOWITZ POINT

X1142 = trilinear product of centers of Malfatti circles
< 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, 179 }: Central Line L(1, 179) 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_179 = 1st AJIMA-MALFATTI POINT (0.5808, 0.2529) X_1142 = 1st MALFATTI-RABINOWITZ POINT (0.5841, 0.2601) A B C X1 X179 X1142 (a, b, c) = (5, 6, 7) | AB = 1

X1142 (1st MALFATTI-RABINOWITZ POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(A,B,C) : f(B,C,A) : f(C,A,B)\), where \(f(A,B,C) = 1 - 2(1 + \cos B/2)(1 + \cos C/2)/(1 + \cos A/2)\) (M. Iliev, 5/13/07)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(1.113199982240\)
▸ Alternate Coordinates1 form

  • Trilinears: \(g(A,B,C) : g(B,C,A) : g(C,A,B)\), where \(g(A,B,C) = 1 - 4 \sec^{2}(A/4)\cos^{2}(B/4)\cos^{2}(C/4)\) (M. Iliev, 5/13/07)

▸ Incident Central Lines1 line

{1, 179}

▸ Functional Relations1 relation

  • X1142 = trilinear product of centers of Malfatti circles

Notes

Let A', B', C' be the respective centers of the three Malfatti circles of ABC. Let A″ be the point of intersection of lines BC' and CB', and define B″ and C″ cyclically. Then triangle A″B″C″ is perspective to ABC, and the perspector is X1142. (Stanley Rabinowitz, #4610, 12/29/01; coordinates by Paul Yiu, #4614, 12/30/01)

Let A', B', C' be the centers of the Malfatti circles. Let A″ be the trilinear product B'*C', and define B″ and C″ cyclically. The lines AA″, BB″, CC″ concur in X1142. The lines A'A″, B'B″, C'C″ concur in X179. (Randy Hutson, March 21, 2019)