Exercise 12. Let $A B C$ be a triangle such that $A B=A C=130$ and $B C=240$. A circle of radius $R$ is tangent to $(A B)$ at $B$ and to $(A C)$ at $C$. Determine the value of $R$.
Solution to Exercise 12 Let $D$ be the point diametrically opposite to $A$ on the circumcircle of triangle $ABC$. We have $(DB) \perp (BA)$ and $(DC) \perp (AC)$. Furthermore, due to symmetry, we have $DB = DC$, so $D$ is the center of the circle with radius $R$ in the statement.

Let $M$ be the midpoint of $[BC]$. Then $ABM$ is a right triangle at $M$ with $BM = 120$ and $AB = 130$, so using the Pythagorean theorem, we find that $AM = 50$.

By writing the tangent of angle $\widehat{BAM}$ in two ways, we get $\frac{DB}{AB} = \frac{BM}{AM}$, so $\frac{R}{130} = \frac{120}{50}$, which simplifies to $R = 312$.