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Published byLaurence Reynolds Modified over 9 years ago
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Geometry
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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
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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?
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Proving that the Circumcircle Exists
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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
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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?
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Deriving the Law of Sines Contd.
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Solution
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Relating Circumradius and Area
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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
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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
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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.
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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?
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Misc. Topics - Angle Bisector Theorem
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Misc. Topics - Power of a Point (POP) R
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