MATH 5200/7200 Practice Problems, 10/17/08

Prove each of the following theorems using only our axioms, basic theorems, or theorems proved in class or homework. Please write your proofs in complete sentences, not in outline form.

1. If ABC is a triangle with altitudes AD, BE, and CF, then (AD)(BC) = (BE)(AC) = (CF)(AB).

2. If two sides of a quadrilateral are equal and parallel, then the quadrilateral is a parallelogram.

3. Suppose that the tangents to a circle C at points A and B intersect at a point P. Then PA = PB, and the ray from P through the center of the circle bisects angle APB.

4. If two circles have two points A and B in common, then the line through the centers of the two circles is the perpendicular bisector of the segment AB.

5. An angle formed by two secants intersecting outside a circle is equal to half the difference of the angles of the two arcs intercepted by the secants.

In other words, in the following diagram if a is the measure of the angle APC, then a = 1/2(x - y), where x is the angle measure of the arc AC, and y is the angle measure of the arc BD.