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 2 of 5: Excircle tangent lengths
Previous stepNext step
In plain words

The excircle sees the supplementary angles, converting cotangents into tangents.

c=q(tan⁡A2+tan⁡B2)c=q\left(\tan\frac A2+\tan\frac B2\right)
Detailed analysis

For the excircle in the angle at CC, the corresponding angles are π−A\pi-A and π−B\pi-B. Thus the same tangent-length argument gives c=q(cot⁡π−A2+cot⁡π−B2)=q(tan⁡A2+tan⁡B2)c=q\left(\cot\frac{\pi-A}{2}+\cot\frac{\pi-B}{2}\right)=q\left(\tan\frac A2+\tan\frac B2\right).