X179

1st AJIMA-MALFATTI POINT

The famous Malfatti Problem is to construct three circles O(A), O(B), O(C) inside ABC such that each is externally tangent to the other two, O(A) is tangent to lines AB and AC, O(B) is tangent to BC and BA, and O(C) is tangent to CA and CB. Let A' = O(B)∩O(C), B' = O(C)∩O(A), C' = O(A)∩O(B). The lines AA',BB',CC' concur in X179. Trilinears are found in Yff’s unpublished notes. See also the Yff-Malfatti Point, X400, having trilinears \(\csc^{4}(A/4)\) : \(\csc^{4}(B/4)\) : \(\csc^{4}(C/4)\), and the references for historical notes.
< 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, 1142 }: Central Line L(1, 1142) 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_1142 = 1st MALFATTI-RABINOWITZ POINT (0.5841, 0.2601) X_179 = 1st AJIMA-MALFATTI POINT (0.5808, 0.2529) A B C X1 X1142 X179 (a, b, c) = (5, 6, 7) | AB = 1

X179 (1st AJIMA-MALFATTI POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(\sec^{4}(A/4) : \sec^{4}(B/4) : \sec^{4}(C/4)\)
Barycentrics: \(\sin A \sec^{4}(A/4) : \sin B \sec^{4}(B/4) : \sin C \sec^{4}(C/4)\)
Search Value (6, 9, 13): \(1.257739557595\)
▸ Incident Central Lines3 lines

{1, 1142}, {485, 8242}, {1143, 9795}

Notes

H. Fukagawa and D. Pedoe, Japanese Temple Geometry Problems (San Gaku), The Charles Babbage Research Centre, Winnipeg, Canada, 1989.

Michael Goldberg, “On the original Malfatti problem,” Mathematics Magazine, 40 (1967) 241-247.

Clark Kimberling and I. G. MacDonald, Problem E 3251 and Solution, American Mathematical Monthly 97 (1990) 612-613.

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, July 11, 2019)

For an artistic design generated by X179, see X244.