Suppose we have a right triangle $ABC$ with hypotenuse $AB$. Further, suppose the altitude from $C$ hits hypotenuse $AB$ at point $D$. If the inradius of $ACD$ is $r_1$ and the inradius of $BCD$ is $r_2$, then what is the inradius of triangle $ABC$, in terms of $r_1$ and $r_2$? My guess is that it is $\sqrt{r_1^2+r_2^2}$, but can someone help me confirm or deny this statement?
Relationship Between Three Incircles
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geometry
euclidean-geometry
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0ahh, it appears I have not done enough research... http://agutie.homestead.com/files/problem/p023_right_triangle_incircles.htm... but how does one arrive at that? Of course, one obvious method is monstrous algebra bash using the relationships between sids of the right triangle and repeatedly utilizing $A=rs$, but I feel there should be a simple proof for such a simple theorem. – 2012-01-26