1.  A line l  said to be tangent to a circle C if l intersects C at exactly one point.   Prove

Theorem:  Given a circle C and a point P lying on C, the line through P perpendicular to the radius is the unique line through P tangent to C.

Note:   You have to prove two things:
 a)   that the line perpendicular to the radius meets the circle only at one point,   and
 b)   no other line through P has this property.

2.  Prove

Theorem:  Given a circle with diameter AB,  and given a third point C on the circle,  the angle ACB is a right angle.

3.  The problem here is to construct the lines tangent  to  a given circle,  passing through a given point outside the circle.

a) Make a GSP sketch of the following construction:

Given:   Circle with center E,  and point A outside the circle.  Construct the circle with diameter AE.   Let P and Q be the points of intersection of this circle with the original circle.   Construct lines AP and AQ.

b)  Prove that  the construction in part a produces two lines tangent to the original circle.   Your proof should include an explanation of why P and Q exist.

4.  Euclid  solves problem 3  a different way,  in Book III,  proposition 17.
Describe Euclid's solution,   and use Side-Angle-Side congruence to prove that it works.

5. State and prove the Pythagorean theorem. (For this problem it is OK to look up a proof in a book or on the internet.) Use GSP to illustrate your proof. Write a complete list of the facts that are used in your proof.
Be prepared to present your proof to the class.