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 3 of 5: A general radius ratio
Previous stepNext step
In plain words

The common base length cancels, leaving a formula for any triangle.

rq=tan⁡A2tan⁡B2\dfrac rq=\tan\dfrac A2\tan\dfrac B2
Detailed analysis

Divide the two expressions for cc. Since 1cot⁡x=tan⁡x\frac{1}{\cot x}=\tan x, cancellation gives rq=tan⁡A2tan⁡B2\frac rq=\tan\frac A2\tan\frac B2.