### 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.

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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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