Homework problems

Problem 1. Prove that the altitudes h_a, h_b and h_c of a triangle and the radii r_a, r_b and r_c of the escribed circles satisfy the equality

$$\frac{1}{h_a}+\frac{1}{h_b}+\frac{1}{h_c}=\frac{1}{r_a}+\frac{1}{r_b}+\frac{1}{r_c}.$$

Hint. See the solution of Problem 4.

Problem 2. Using the notation above, prove that

$$AI_a=\frac{r_a}{\sin \frac{1}{2} \angle A}.$$

Hint. Use that the triangles I_a I_b I_c and I_a BC are similar and problem 3.