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Chapter 13: The concurrence Theorems

Chapter 13: The concurrence Theorems. By: M itch Maslanka. Lesson 1: Triangles and Circles Definitions. A polygon is cyclic iff there exists a circle that contains all of its vertices. Circumcircle- a circle which passes through all of the vertices of a polygon.

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Chapter 13: The concurrence Theorems

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  1. Chapter 13:The concurrence Theorems By: Mitch Maslanka

  2. Lesson 1: Triangles and CirclesDefinitions • A polygon is cyclic iff there exists a circle that contains all of its vertices. • Circumcircle- a circle which passes through all of the vertices of a polygon. • Circumcenter-the center of the circle circumscribed about the polygon.

  3. Lesson 1: Triangles and CirclesDefinitions • A polygon is inscribed in a circle iff each vertex of the polygon lies on the circle. • The circle is circumscribed about the polygon.

  4. Lesson 1: Triangles and CirclesTheorems • Theorem 68: Every triangle is cyclic • Corollary to theorem 68: The perpendicular bisectors of the sides of a triangle are congruent.

  5. Lesson 2: Cyclic QuadrilateralsTheorems • Theorem 69: A quadrilateral is cyclic iff a pair of its opposite angles are supplementary.

  6. Lesson 3: IncirclesDefinitions • A circle is inscribed in a polygon iff each side of the polygon is tangent to the circle. • The polygon is circumscribed about the circle. • The circle is called the incircle of the polygon, and its center is called the incenter of the polygon.

  7. Lesson 3: IncirclesTheorems • Theorem 70: Every triangle has an incircle. • Corollary to theorem 70: The angle bisectors of a triangle are concurrent.

  8. Lesson 4: The Centroid of a TriangleMedians and Centroids • A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side. • Theorem 71: The medians of a triangle are concurrent. • The centroid of a triangle is the point in which its medians are concurrent

  9. Lesson 4: The Centroid of a TriangleAltitudes and Orthocenters • Theorem 72: The lines containing the altitudes of a triangle are concurrent. • The orthocenter of a triangle is the point in which the lines containing its altitudes are concurrent.

  10. Chart

  11. Lesson 5: Ceva’s Theorem • A cevian of a triangle is a line segment that joins a vertex of the triangle to a point on the opposite side. (Angle bisectors, medians, and altitude) • Theorem 73. Ceva’s Theorem: Three cevians, AY, BZ, and CX, of ∆ABC are concurrent iff x A X Z B C Y

  12. Lesson 6: Napoleon’s Discovery and Other Surprises • Slope: m= • Distance formula: d= • Know how to graph

  13. Review Questions • A polygon is (blank) iff there exists a circle that contains all of its vertices. • The (blank) of a triangle is the point in which its medians are concurrent.

  14. Review Questions (cont.) 3. Find x 4. Quadrilateral EFGH is a rhombus in which <E=112°. Is EFGH cyclic? B 3 5 8 4 A C x 6

  15. Answers • Cyclic • Centroid • X=1.8 • No, 112 +112 doesn’t equal 180.

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