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: Apply the ratio to the two subtriangles

Previous step Next step

In plain words

Cutting the triangle creates two smaller triangles, but the same radius relation remains valid.

\frac{r_1}{q_1}=\tan\frac A2\tan\frac{\angle AMC}{2},\quad\frac{r_2}{q_2}=\tan\frac B2\tan\frac{\angle BMC}{2}

Detailed analysis

Apply the general formula to $AMC$ and $BMC$ . Their angles adjacent to $C$ are $A$ and $B$ , while their other relevant angles are $\angle AMC$ and $\angle BMC$ .