Homework problems

Problem 1. Prove that the circumcircle of $\Delta ABC$ is the nine-point circle of $\Delta I_a I_b I_c$, where I_a, I_b and I_c are the centers of the excircles of $\Delta ABC$.

Problem 2. Prove using Ceva's theorem that altitudes of a given triangle are concurrent.

Problem 3. Prove that the external bisectors of the three angles of a given triangle meet their respective opposite sides at three collinear points.