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Proving Centers of Triangles Adapted from Walch Education

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Circumcenter The perpendicular bisector is the line that is constructed through the midpoint of a segment. The three perpendicular bisectors of a triangle are concurrent, or intersect at one point. This point of concurrency is called the circumcenter of the triangle. The circumcenter of a triangle is equidistant, or the same distance, from the vertices of the triangle. This is known as the Circumcenter Theorem. 1.9.4: Proving Centers of Triangles2

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3 Theorem Circumcenter Theorem The circumcenter of a triangle is equidistant from the vertices of a triangle. The circumcenter of this triangle is at X.

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Circumcenter, continued The circumcenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle. The circumcenter is inside acute triangles, outside obtuse triangles, and on the midpoint of the hypotenuse of right triangles. 1.9.4: Proving Centers of Triangles4

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Circumcenter, continued 1.9.4: Proving Centers of Triangles5 Acute triangleObtuse triangleRight triangle X is inside the triangle. X is outside the triangle. X is on the midpoint of the hypotenuse.

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Circumcenter, continued The circumcenter of a triangle is also the center of the circle that connects each of the vertices of a triangle. This is known as the circle that circumscribes the triangle. 1.9.4: Proving Centers of Triangles6

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Incenter The angle bisectors of a triangle are rays that cut the measure of each vertex in half. The three angle bisectors of a triangle are also concurrent. This point of concurrency is called the incenter of the triangle. The incenter of a triangle is equidistant from the sides of the triangle. This is known as the Incenter Theorem. 1.9.4: Proving Centers of Triangles7

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8 Theorem Incenter Theorem The incenter of a triangle is equidistant from the sides of a triangle. The incenter of this triangle is at X.

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Incenter, continued The incenter is always inside the triangle. 1.9.4: Proving Centers of Triangles9 Acute triangleObtuse triangleRight triangle

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Incenter, continued The incenter of a triangle is the center of the circle that connects each of the sides of a triangle. This is known as the circle that inscribes the triangle. 1.9.4: Proving Centers of Triangles10

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Orthocenter The altitudes of a triangle are the perpendicular lines from each vertex of the triangle to its opposite side, also called the height of the triangle. The three altitudes of a triangle are also concurrent. This point of concurrency is called the orthocenter of the triangle. 1.9.4: Proving Centers of Triangles11

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Orthocenter, continued The orthocenter can be inside the triangle, outside the triangle, or even on the triangle depending on the type of triangle. The orthocenter is inside acute triangles, outside obtuse triangles, and at the vertex of the right angle of right triangles. 1.9.4: Proving Centers of Triangles12

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Orthocenter, continued 1.9.4: Proving Centers of Triangles13 Acute triangleObtuse triangleRight triangle X is inside the triangle. X is outside the triangle. X is at the vertex of the right angle.

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Centroid The medians of a triangle are segments that join the vertices of the triangle to the midpoint of the opposite sides. Every triangle has three medians. The three medians of a triangle are also concurrent. This point of concurrency is called the centroid of the triangle. The centroid is always located inside the triangle the distance from each vertex to the midpoint of the opposite side. This is known as the Centroid Theorem. 1.9.4: Proving Centers of Triangles14

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1.9.4: Proving Centers of Triangles15 Theorem Centroid Theorem The centroid of a triangle is the distance from each vertex to the midpoint of the opposite side. The centroid of this triangle is at point X.

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Centroid, continued The centroid is always located inside the triangle. The centroid is also called the center of gravity of a triangle because the triangle will always balance at this point. 1.9.4: Proving Centers of Triangles16 Acute triangleObtuse triangleRight triangle

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Point of Concurrency Each center serves its own purpose in design, planning, and construction. 1.9.4: Proving Centers of Triangles17 Center of triangleIntersection of CircumcenterPerpendicular bisectors IncenterAngle bisectors OrthocenterAltitudes CentroidMedians

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