Coordinate Proofs with Circles

You can prove many theorems about circles by using coordinate geometry. Whenever possible locate the circle so that its center is at the origin.

1. Prove that an angle inscribed in a semicircle is a right angle. Use the figure. (Hint: Write an equation for the circle. Use your equation to help show that slope of AP * slope of PB = -1.

2. Suppose PQ is perpendicular to AB, Q is between A and B, and PQ is the geometric mean between AQ and QB. Prove that P is on the circle that has AB as a diameter. Use the figure below.