Journey to the Center of a
Triangle:Where are the four centers?
The four centers of a triangle fall in different places, depending
on what kind of triangle it is. In your explorations of the centers,
you probably already noticed this. Sometimes a particular "center"
will fall in a very surprising place - such as outside the triangle!
You may wonder, how could we call this type of point a "center" of
the triangle? Well, again, it depends on what we mean by center! For
example, in an obtuse triangle, as you will see below, the
circumcenter falls outside the triangle, and not in the interior at
all. Yet it is still equidistant from each of the vertices of the
triangle, it is still the center of a circumscribed circle, and it is
still the Circumcenter!
In each diagram below, the points are labelled in the following
way: O=orthocenter, I=incenter,
C=circumcenter and G=centroid:
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In an acute triangle, the 4 centers are all inside the
triangle:
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For a right triangle, the Orthocenter is on the right
angle vertex, the Circumcenter is the midpoint of
hypotenuse, and the other 2 centers are somewhere inside the
triangle.
This relates directly to the properties of inscribed
angles in circles: an inscribed angle is equal to one-half
its intercepted arc, and in a right triangle, the right
angle is equal to 90 degrees and therefore the intercepted
arc is equal to 180 degrees, a semi-crcle!
An additional interesting fact: we should not be
surprised to see that the Circumcenter falls on the midpoint
of the hypotenuse, because the midpoint of the hypotenuse of
a right triangle is equidistant from the 3 vertices of the
triangle. (Recall that 2 right triangles make a rectangle,
and the diagonals or a rectangle are equal and bisect each
other.
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In an obtuse triangle, the Circumcenter and Orthocenter
are outside the triangle, while the other 2 centers are
inside the triangle.
This, too, relates directly to the properties of
inscribed angles in circles: an inscribed angle is equal to
one-half its intercepted arc, and in an obtuse triangle, the
angle is greater than 90 degrees and therefore the
intercepted arc is greater than 180 degrees!
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For a scalene triangle, the 4 centers could be anywhere inside or
outside triangle, depending on whether the triangle is acute, right
or obtuse.
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In an isosceles triangle, the 4 centers are collinear.
This is true because in an isosceles triangle, the altitude
to the base is also a median and an angle bisector. This
line is the one line of reflection symmetry for the
triangle.
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In an equilateral triangle, all centers are the same
point. This is because the altitude to each side of an
equilateral triangle is also a median and an angle bisector.
The equilateral triangle is the most symmetrical triangle of
all, and has 3 lines of reflection symmetry (those same
altitudes). The equilateral triangle has 120 degree rotation
symmetry about the "center" (Orthocenter, Centroid,
Circumcenter, and Incenter). The equilateral triangle is the
only triangle that does have rotation symmetry.
This point is truly the center of the
triangle.
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Foreward to Applications of
the Center of a Triangle
Back to the Introduction to Journey to the
Center of a Triangle