Homework problems

Problem 1. Prove that the sum of the diagonals of a convex pentagon is greater than the perimeter but less than double the perimeter

Hint. Compare with Problem 7.

Problem 2. Prove that the length of median AM in triangle ABC is not greater than half the sum of sides AB and AC. Prove also that the sum of the lengths of the three medians is not greater than the triangle's perimeter.

Hint. Construct Parallelogram ABDC from triangle ABC.

Problem 3. Prove that a convex pentagon (that is, a pentagon whose diagonals all lie inside the figure) has three diagonals which can form a triangle

Hint. Consider the longest diagonal XY of the pentagon. There exist two intersecting diagonals, each of which has X and Y as one endpoint respectively. Prove that these three diagonals satisfies the three triangle inequalities.