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: Incircle tangent lengths
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In plain words

The base is split into two tangent lengths.

c=r(cot⁡A2+cot⁡B2)c=r\left(\cot\frac A2+\cot\frac B2\right)
Detailed analysis

Let the incircle touch ABAB at TT. The right triangles at AA and BB give AT=rcot⁡A2AT=r\cot\frac A2 and BT=rcot⁡B2BT=r\cot\frac B2; adding them yields the displayed identity.