1. Geometry

2. Circumcircle of a Triangle For any triangle, there is a unique circle that is tangent to all three vertices of the triangle This circle is the circumcircle of said triangle

3. Proving that the Circumcircle Exists How can we show that every triangle has a circumcircle? Think about the properties of a circumcircle’s center, or the circumcenter – what is its relationship with the vertices of its inscribed triangle?

4. Proving that the Circumcircle Exists

5. Law of Sines We can use the circumcircle of triangle ABC to come up with a stronger version of the law of sines involving the circumradius, r, of triangle ABC

6. Deriving the Law of Sines To get you started, here’s the first step: draw the circumdiameter through any of the vertices of ABC, as shown below. Can you use this diagram to relate Sin C, side c, and the circumdiameter?

7. Deriving the Law of Sines Contd.

8. Solution

9. Relating Circumradius and Area

10. Incircles Just as every triangle has a circumcircle, every triangle also has an incircle that’s internally tangent to each of the triangle’s sides

11. Proving that the Incircle Exists We can employ a tactic similar to the one we used for the circumcircle Look for a geometric figure that contains all of the points equidistant from two sides

12. Relating the Inradius and Area We can derive a formula relating inradius, area, and semiperimeter by using the fact that the incircle is tangent to each side of a triangle.

13. Competition Problem Try relating area, inradius, and semiperimeter to solve the following problem (2012 AMC 12A # 18) Triangle ABC has AB = 27, AC = 26, and BC = 25. Let I denote the intersection of the internal angle bisectors of ABC. What is BI?

14. Misc. Topics - Angle Bisector Theorem

15. Misc. Topics - Power of a Point (POP) R