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# Congruent Incircles

Given a triangle, draw a Cevian to one of the bases that divides it into two triangles having congruent incircles. The positions and sizes of these two circumcircles can then be determined by simultaneously solving the eight equations

 (1) (2) (3) (4) (5) (6) (7) (8)

for the eight variables , , , , , , , and , with , , and given. Generalizing to congruent circles gives the equations

 (9) (10) (11)

for , ..., ,

 (12)

for , ..., , and

 (13)

to be solved for the unknowns and ( of them), and ( of each for , ..., ), and , , , and , a total of unknowns.

Given an arbitrary triangle, let Cevians be drawn from one of its vertices so all of the triangles so determined have equal incircles. Then the incircles determined by spanning 2, 3, ..., adjacent triangles are also equal (Wells 1991, p. 67).

Congruent Incircles Point, Incircle

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## References

Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, 1991.

## Referenced on Wolfram|Alpha

Congruent Incircles

## Cite this as:

Weisstein, Eric W. "Congruent Incircles." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CongruentIncircles.html