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.
Information from Kimberling's Encyclopedia of Triangle Centers |
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)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 Trilinears are found in Yff's unpublished notes.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 csc^{4}(A/4) : csc^{4}(B/4) : csc^{4}(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