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 5 of 5: Conclude by Stewart’s theorem
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In plain words

A standard cevian theorem closes the second route.

Stewart’s theorem holds, hence r1q1r2q2=rq\text{Stewart’s theorem holds, hence }\dfrac{r_1}{q_1}\dfrac{r_2}{q_2}=\dfrac rq
Detailed analysis

Stewart’s theorem holds for every triangle and every cevian. Therefore the algebraic identity, and hence the original radius identity, is true.