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: Translate the target into algebra
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In plain words

A geometry statement has become a length identity.

b+d−c1b+d+c1a+d−c2a+d+c2=a+b−ca+b+c\frac{b+d-c_1}{b+d+c_1}\frac{a+d-c_2}{a+d+c_2}=\frac{a+b-c}{a+b+c}
Detailed analysis

Substitute the two subtriangle ratios and the ratio for ABCABC. The desired geometric equality is now an equality involving only a,b,c,c1,c2,da,b,c,c_1,c_2,d, with c=c1+c2c=c_1+c_2.