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JapaneseTempleProblem

The positions of points *A*, *B*, *C*, *D*, and *E* are identical on the two large circles. Consequently, the polygons *ABCDE* and *A'B'C'D'E'* are congruent. However, in one polygon, the interior triangles share point *A*, but in the other, they share point *B'*, so that the two triangulations are not identical.
According to one of many problems carved in wood and hung in a Japanese temple 200 years ago, "to the glory of the gods and the discoverer", the sums of inradii are equal. You see here a confirmation for the case of three interior circles, but the theorem generalizes to any number of such circles.

For more, see *GEOMETRY IN ACTION*, Chapter 7, Project 3.