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An orthocenter of a
triangle is the point where the three altitudes of a
triangle meet. This point does not necessarily lie
inside the triangle. The most remarkable theorem
states the the circumcenter, the centroid, and the
orthocenter all lie on the same line, which is called
the Euler line.
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Basic
Altitude Theorems
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1) The three altitudes of a triangle all
meet at a point.
2) The orthocenter of a triangle is the
circumcenter of its anticomplementary
triangle.
3) The circumcenter of a triangle is the
orthocenter of its complementary
triangle.
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Euler Line
Theorem
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In any triangle, the circumcenter (
O),
the centroid (G), and the orthocenter (H) all
lie on a straight line with the centroid
between the circumcenter and the
orthocenter.
Furthermore, the distance between the
centroid and the orthocenter is twice the
distance between the centroid and the
circumcenter.
HG = 2 OG
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The
Famous Nine-Point Circle
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Above image was done using the
9-Point Circle Script in Sketchpad.
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Given any triangle, there exists a circle
whose center is the midpoint of the Euler
line and radius is 1/2 the circumradius of
the circle.
The circle has 9 points on it:
a) the three midpoints of the
sides of the triangle:
m1, m2, and m3,
b) the points of intersection of the
altitudes of the triangle with their
respective sides:
p1, p2, and p3,
3) the midpoints of each segment from
the orthocenter to a vertex of the
triangle:
a1, a2, and a3.
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