1. Lecture 07: Geometry of the Circle
Hans Li, Wilbur Li

2. Triangles Review!

3. Cyclic Quadrilaterals
Equal inscribed angles
Supplementary Opposite Angles
Intersecting right angles
Great for angle chasing and showing two angles are supplementary!

4. Ptolemy’s Theorem If quadrilateral ABCD is cyclic, then:
AB · CD + BC · DA = AC · BD
Challenge! Prove Ptolemy’s Theorem.
Hint: Use the diagram to the left. Find similar triangles that relate the sides in Ptolemy’s equation.

5. Power of a Point

6. Other Methods
Similarity
Congruency
Inscribed Angles
Draw a picture!

7. Big Boy Time: IMO 1979 :)
In triangle ABC we have AB = AC. A circle that is internally tangent with the circumscribed circle is also tangent to the sides AB, AC at the points P, Q, respectively. Prove that the midpoint of PQ is the center of the inscribed circle of the triangle ABC.