MathLabs

Problem 1

Let MM be a point on side ABAB of triangle ABCABC. Let r1,r2,rr_1,r_2,r be the inradii of triangles AMCAMC, BMCBMC, and ABCABC, respectively. Let q1,q2,qq_1,q_2,q be the radii of the excircles of these triangles lying in angle ∠ACB\angle ACB. Prove that r1q1⋅r2q2=rq\frac{r_1}{q_1}\cdot\frac{r_2}{q_2}=\frac rq.
Step 1 of 5: Radius ratio from the angle bisector
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In plain words

Both circles touch the two sides through CC, so their scales are controlled by the same angle bisector.

rq=s−cs=a+b−ca+b+c\dfrac rq=\dfrac{s-c}{s}=\dfrac{a+b-c}{a+b+c}
Detailed analysis

The incenter and the CC-excenter lie on the bisector of ∠C\angle C. Similar right triangles show that their radius ratio equals the ratio of the tangent lengths from CC, namely s−cs\frac{s-c}{s}.