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PROPERTIES OF THE ISOSCELES TRIANGLE
Trigonometric and Geometric Calculations 195
You have already seen that a right triangle is a useful building block for other shapes.
An isosceles triangle has slightly different uses. An isosceles triangle has two equal sides
and two equal angles opposite those two sides. A perpendicular from the third angle
(not one of the equal angles) to the third side (not one of the equal sides) bisects that
third side. That is, it divides it into two equal parts, making the whole triangle into
mirror-image right triangles, as shown in Fig. 14-7.
Any triangle except a right triangle can be divided into three adjoining isosceles triangles
by dividing each side into two equal parts and erecting perpendiculars from the
points of bisection. Where any two of these bisecting perpendiculars meet, if lines are
drawn to the corners of the original triangle, the three lines have equal lengths. That is
so because two of them form the sides of an isosceles triangle. So the perpendicular
from the third side of the original triangle must also meet in the same point.
This statement is true whether the original triangle is acute or obtuse. Figure 14-8 shows
an example with an acute triangle, where the meeting point is inside. In the case of an
obtuse triangle, the meeting point still exists, but it is outside the triangle rather than
inside it.
If you try to apply this rule to a right triangle, perpendiculars from the midpoint of the
hypotenuse to the other two sides bisect those two sides. But the meeting point of those
perpendiculars lies on the hypotenuse, so the third isosceles triangle disappears.
Figure 14-7
Properties of an
isosceles triangle.
The two equal
angles are opposite
the two equal
sides.