2. Suppose that $\mathrm{\Delta}ABC$ is equilateral triangle and that P is a point in the interior of this triangle. Prove that the sum of the perpendicular distances from P to each of the sides of the triangle is equal to the height of the triangle.

3. Prove that the area of a triangle is equal to one-half the product of its perimeter and the length of the radius of a circle inscribed within the triangle.

4. Prove that the diagonal of a kite connecting the vertices where the congruent sides intersect bisects the angles at these vertices and is the perpendicular bisector of the other diagonal. (Use the strict definition of a kite which is a quadrilateral with two distinct pairs of congruent adjacent sides.)