Problem 1 (25 points)

Given two lines m and n that intersect at O and a point P distinct from O, prove that P is equidistant from m and n if and only if OP bisects one of the four angles formed by the intersecting lines m and n.

Problem 2 (20 points)

Construct with straightedge and compass a circle that is tangent to the segment BC and tangent to the rays BA and CD.  Label key points and write down the MAJOR STEPS of your construction.  (Do not get carried away; write down 3 or 4 steps so that one can tell how the key points, lines and circles in your construction were made.)

Problem 3 (20 points)

Tools:  In this problem you can use any theorem or homework problem proved in the course.

Let ABCD be a quadrilateral inscribed in a circle.  If |AB| = |CD|, prove that BC is parallel to DA.

Problem 4 (35 points)

(4a. 15 points). Given a point D on segment AB below, construct (with straightedge and compass) a point C so that ABC is a right triangle with right angle ACB and segment CD is an altitude.

(4b. 15 points). Suppose the lengths of the segments in the construction problem above were |AD|=3 and |BD| = 12, what would be the length of the altitude |CD| in right triangle ABC?  Show your reasoning.  You can use any theorems we have proved.  Note:  These are not the lengths above; this is not a measurement problem but a reasoning problem.

(4c. 5 points). With the lengths given in (b), the point D can be expressed as an affine combination (center of mass), D = sA + tB, where s and t are masses at A and D that make D the center of mass. Tell what are the numbers s and t in this case.  (A numerical answer is all that is called for, not an explanation.)