### Basic Geometry Definitions

The Centroid, G, of a triangle is the common intersection of three meedians. A median of a triangle is the segment from a vertex to the midpoint of the side of the triangle opposite that vertex.

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The Orthocenter H of a triangle is the common intersection of the three lines containing the altitudes. An altitude is a perpendicular segment from a vertex to the side opposite.

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Let Triangle ABC be any triangle. Then if P is any point in the plane. Constuct perpendiculars to the sides of ABC (extended if necassary). Let R,S, T be the three points where the the perpendiculars and the sides (or extended sides) of ABC meet. RST is the Pedal Triangle defined by P, the Pedal Point, and triangle ABC.

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The Circumcenter, C, of a triangle is the point in the plane equidistant from the three vertices of the triangle. Since a point equidistant from two points lies on the perpendicular bisector determined by the of the two points,C, is on the perpendicular bisector of each side the triangle. C may be outside the triangle.

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The Circumcircle of a triangle has center at the circumcenter of the triangle and all three vertices are on the circle.

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The Incenter,I, of a triangle is the point on the interior of the triangle that is equidistant from the three sides. Since a point interior to an angle that is equidistant from the two sides of the angle lies on the angle bisector, then I must be on the angle bisector of each angle of the triangle.

The Medial Triangle of a triangle ABC is the triangle constructed by connecting the three midpoints of triangle ABC. It is similar to ABC and one-fourth of its area.

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The Orthhic triangle of a given acute triangle is the triangle constructed by connecting the feet of the altitudes of the given triangle.

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The Nine Point Circle of a given triangle is the circle that passes through the three mid-pointss of the sides, the three feet of the altitudes, and the three midpoints of the segments from the respective vertices to the orthocenter.

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The Euler line of a triangle is the segment containing H,G,C of the triangle

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