## X(179) (1ST AJIMA-MALFATTI POINT)

 Interactive Applet

You can move the points A, B and C (click on the point and drag it).
Press the keys “+” and “−” to zoom in or zoom out the visualization window and use the arrow keys to translate it.

You can also construct all centers related with this one (as described in ETC) using the “Run Macro Tool”. To do this, click on the icon , select the center name from the list and, then, click on the vertices A, B and C successively.

 The JRE (Java Runtime Environment) is not enabled in your browser!

Download all construction files and macros: tc.zip (10.1 Mb).
This applet was built with the free and multiplatform dynamic geometry software C.a.R..

 Information from Kimberling's Encyclopedia of Triangle Centers

Trilinears           sec4(A/4) : sec4(B/4) : sec4(C/4)
Barycentrics    sin A sec4(A/4) : sin B sec4(B/4) : sin C sec4(C/4)

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 X(179). Trilinears are found in Yff's unpublished notes. See also the Yff-Malfatti Point, X(400), having trilinears csc4(A/4) : csc4(B/4) : csc4(C/4), and the references for historical 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.

X(179) lies on this line: 1,1142

This is a joint work of
Humberto José Bortolossi, Lis Ingrid Roque Lopes Custódio and Suely Machado Meireles Dias.

If you have questions or suggestions, please, contact us using the e-mail presented here.

Departamento de Matemática Aplicada -- Instituto de Matemática -- Universidade Federal Fluminense