**Due Wednesday, September 26, 2018**

1. Let S denote the set of all points that are equidistant from two distinct points A and B. Prove that S is the straight line that is perpendicular to the line AB and bisects it (the *perpendicular bisector* of AB).

2. Prove that the perpendicular bisectors of the three sides of any triangle are concurrent (i.e. there is a single point at which all three of these perpendicular bisectors meet).

3. Prove that the diagonals of any parallelogram bisect each other.

Hints:

For problem 1:

Let me start by giving the following definition:

Definition: The *perpendicular bisector* of the straight line AB is the infinite straight line which meets AB at the midpoint of AB, and is perpendicular to AB. Every finite line AB has a perpendicular bisector (this is easy to prove using Propositions 10 and 11 -- you are not required to include a proof of this fact in your HW).

Problem 1 asks you to prove that one set (S) "is" another (the perpendicular bisector of AB).
Proving that two sets are equal generally requires you to write two proofs:
prove that each element of the first set is in the second, and that each
element of the second is in the first. Applying this principle to the current
problem, you should write proofs of the following two statements:

Given a straight line AB:

(i) For any given point C, prove that if CA = CB,
then C is on the perpendicular bisector of AB.

(ii) For any given point C, prove that if C is on the
perpendicular bisector of AB, then CA = CB.

Note (i) and (ii) should be two completely separate proofs. Neither should depend in any way on the other.

For problem 2: Problem 1 might be useful!

For problem 3: Use the following definition of parallelogram:
A *parallelogram* ABCD is a quadrilateral in which AB is parallel to CD and
BC is parallel to AD. (You are not allowed to assume that opposite sides of a parallelogram are equal. If you wish to use this fact, you must first prove it!)