Geometry

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

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?

Proving that the Circumcircle Exists

• Some hints: the distance from the circumcenter (F) to each vertex must be equal. What happens if we show that and ?

• Is there only one point that is equally distant from two other points? Can you find a geometric figure that contains all of the points that are equidistant from a pair of points?

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

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?

Deriving the Law of Sines Contd.

• Here are some more hints: How are angle ADB and angle C related? Think about finding right triangles to use the ratio

Solution

=

Is there anything special about angle C and side c that allowed us to derive the above equation?

Relating Circumradius and Area

• Using the formula we can substitute in to get

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

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

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.

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?

Misc. Topics - Angle Bisector Theorem

• If M is the point at which the angle bisector of angle B intersects then

• Prove the angle bisector theorem using the law of sines

Misc. Topics - Power of a Point (POP) is constant for any line drawn through P for a certain circle R.E.g. in the diagram, =

Prove POP (What are the different cases?)

R