JapaneseTempleProblemThe 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.