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A centroid of a triangle
is its center of gravity, which means that a triangle
can be balanced at the end of a pencil at its
centroid. The centroid is found at the intersection of
the medians, which are the line segments connecting
the three vertices with the midpoints of the
respective opposite sides.
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Basic
Median Theorems
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2) If G is the point of intersection of
the two medians for B and C respectively,
then GB' = 1/3 BB'
OR
G trisects a median.
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3) The three medians of any given triangle
meet at a common point, called the centroid
or center of gravity.
4) The centroid trisects each of the
medians.
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Length
Theorem
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The major theorem about the lengths of the
medians of any triangle is:
2ma2
= b2 + c2 - 1/2
a2
OR
2mb2
= a2 + c2 - 1/2
b2
OR
2mc2
= b2 + c2 - 1/2
c2
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Complementary and Anticomplemetary
Triangles
Given any triangle ABC, then the
complementary (or medial) triangle A'B'C'
is the triangle whose vertices are the
midpoints of the original triangle ABC.
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Conversely, given any triangle
ABC, then the anticomplementary triangle
A"B"C" with the properties that:
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The basic relationship
is:
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