NAME ___________________________

Instructions:  Do all problems.  You will need straightedge and compass.

Problem 1:

Prove that the perpendicular bisectors of the sides of a triangle ABC are concurrent.

Note: You can assume the basic axioms and theorems (except for theorems that are restatements of this theorem, e.g., the existence of a certain circle).

Problem 2:

Let ABC be a triangle with right angle at C; point D is the foot of the altitude through C.  If the lengths of the sides of ABC are a = |BC|, b = |CA|, c = |AB|, find the lengths of AD and CD in terms of a, b, and/or c.  Show your reasoning.

Note:  You can use any theorems.

Problem 3:

Prove that a quadrilateral EFGH is a parallelogram if and only if its opposite angles are congruent (i.e., angle E = angle G and angle F = angle H)..

Note:  You can use any theorems (including theorems about parallel lines) except theorems about parallelograms (which you must prove as part of your proof).

Problem 4:

Construction:  Given the triangle ABC in the figure, construct a circle tangent to all three sides (i.e., inscribed in the triangle).

Problem 5:

Construction:  Given the segment DE in the figure, construct a point F so that DF/DE = 5/7.