An isosceles triangle is a triangle with (at least) two equal sides. In the figure above, the two equal sides have length
and the remaining side has length . This property is equivalent to two angles of the
triangle being equal. An isosceles triangle therefore has both two equal sides and
two equal angles. The name derives from the Greek iso (same) and skelos
(leg).
A triangle with all sides equal is called an equilateral triangle, and a triangle with no sides equal is
called a scalene triangle.
An equilateral triangle
is therefore a special case of an isosceles triangle having not just two, but
all three sides and angles equal. Another special case of an isosceles triangle
is the isosceles right triangle.
The height of the isosceles triangle illustrated above can be found from the Pythagorean theorem as
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(1)
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The area is therefore given by
The inradius of an isosceles triangle
is given by
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(5)
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The mean of is given by
so the geometric centroid
is
or 2/3 the way from its vertex (Gearhart and Schulz 1990).
Considering the angle at the apex of the triangle and writing instead of , there is a surprisingly simple relationship between
the area and vertex angle . As shown in
the above diagram, simple trigonometry
gives
so the area is
Erecting similar isosceles triangles on the edges of an initial triangle gives another
triangle such that , , and concur. The
triangles are therefore perspective
triangles.
No set of points in the plane can determine only isosceles triangles.
Gearhart, W. B. and Schulz, H. S. "The Function ." College
Math. J. 21, 90-99, 1990.
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