Homework problems

Problem 1. Prove that the points where the extended altitudes meet the circumcircle form a triangle similar to the orthic triangle. (The orthic is the triangle formed by the feet of its altitudes)
Problem 2. According to Figure 4 find the point on the circle which has CA as its Simson line.

Problem 3. Proof the following generalization of Ptolemy's theorem. If ABC is a triangle and P is an arbitrary point, then

$$AB.CP+BC.AP \geq AC.BP.$$

In which case the equality sign holds.

Hint. Use the triangle inequality for the pedal triangle for point P.