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Properties of Concurrency in Special Segments of Triangles

Properties of Concurrency in Special Segments of Triangles. Concurrent: Point of Concurrency: Concurrency of Perpendicular Bisectors Theorem: (Theorem 5-6) Circumcenter of the triangle:. When three or more lines intersect at one point, they are concurrent.

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Properties of Concurrency in Special Segments of Triangles

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  1. Properties of Concurrency in Special Segments of Triangles

  2. Concurrent: Point of Concurrency: Concurrency of Perpendicular Bisectors Theorem: (Theorem 5-6) Circumcenter of the triangle: When three or more lines intersect at one point, they are concurrent. The point at which three or more lines intersect is the point of concurrency. The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. The point of concurrency of the perpendicular bisectors of a triangle is the circumcenter of the triangle.

  3. “circumscribed about” Since the circumcenter is equidistant from the vertices, you can use the circumcenter as the center of the circle that contains each vertex of the triangle. You say the circle is circumscribed about the triangle.

  4. Concurrency of Angle Bisectors Theorem: (Theorem 5-6) Incenter of a triangle: The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the angle. The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. “inscribed in”: For any triangle, the incenter is always inside the triangle. In the diagram, points X, Y, and Z are equidistant from P, the incenter of triangle ABC. P is the center of the circle inscribed in the triangle.

  5. Concurrency of Medians Theorem: Centroid of a triangle: The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. In a triangle, the point of concurrency of the medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance. For any triangle, the centroid is always inside the triangle.

  6. Concurrency of Altitudes Theorem: (Theorem 5-9) Orthocenter of the triangle: The lines that contain the altitudes of a triangle are concurrent. The lines that contain the altitudes of a triangle are concurrent at the orthocenter of the triangle. The orthocenter can be inside, on, or outside the triangle.

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