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 4 of 5: Recognize Stewart’s theorem

In plain words

Stewart’s theorem is the hidden conservation law for a cevian.

d^2c+c_1c_2c=a^2c_1+b^2c_2

Detailed analysis

Clear denominators and expand. The resulting equation is exactly Stewart’s theorem for cevian $CM=d$ splitting $AB=c$ into $c_1$ and $c_2$ .