geometry. circumcircle of a triangle for any triangle, there is a unique circle that is tangent to...
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![Page 1: 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](https://reader035.vdocuments.net/reader035/viewer/2022062803/56649f045503460f94c18bef/html5/thumbnails/1.jpg)
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
• 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?
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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.
• Here are some more hints: How are angle ADB and angle C related? Think about finding right triangles to use the ratio
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Solution
=
Is there anything special about angle C and side c that allowed us to derive the above equation?
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Relating Circumradius and Area
• Using the formula we can substitute in to get
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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
• 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
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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