Problem 1

Let

M

be a point on side

AB

of triangle

ABC

. Let

r_1,r_2,r

be the inradii of triangles

AMC

BMC

, and

ABC

, respectively. Let

q_1,q_2,q

be the radii of the excircles of these triangles lying in angle

\angle ACB

. Prove that

\frac{r_1}{q_1}\cdot\frac{r_2}{q_2}=\frac rq

Step 2 of 5: Write the two subtriangle ratios

In plain words

Replace each radius by side lengths; the circles disappear from the algebra.

\frac{r_1}{q_1}=\frac{b+d-c_1}{b+d+c_1},\quad\frac{r_2}{q_2}=\frac{a+d-c_2}{a+d+c_2}

Detailed analysis

Set $AM=c_1$ , $MB=c_2$ , $CM=d$ , $BC=a$ , $CA=b$ , and $AB=c=c_1+c_2$ . Apply the preceding formula to $AMC$ and $BMC$ .