Lecture 07: Geometry of the Circle

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Presentation transcript:

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

Triangles Review!

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

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.

Power of a Point

Other Methods Similarity Congruency Inscribed Angles Draw a picture!

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.