22 II. ON TRIANGLES
Figure 19b
The equilateral triangle, which has all three sides equal;
The right triangle, which has a right angle. The side opposite the right angle
is called the
hypotenuse.1
A perpendicular dropped from a vertex of a triangle onto the opposite side is
called an altitude of the triangle; a median is a line which joins a vertex with the
midpoint of the opposite side.
23. Theorem. In an isosceles triangle, the angles opposite the equal sides are
equal.
Figure 20
Suppose ABC is an isosceles triangle (Fig. 20). Let us turn angle BAC onto
itself (10) so that AB lies along the line of AC and vice-versa. Since AB and AC
are equal, point B will fall on C, and C on B. Angle ABC will therefore fall on
ACB, so that these angles are equal. QED
Converse. If two angles of a triangle are equal, the triangle is isosceles.
In triangle ABC, suppose B = C. Let us move this triangle so that side BC
is turned around, switching points B and C. Since ABC = ACB, side BA will
assume the direction of of CA, and conversely. Point A, which is the intersection
of BA and CA, will therefore retain its original position, so that segment AB falls
on segment AC. QED
Corollary. An equilateral triangle is also equiangular (that is, its three angles
are equal), and conversely.
1The
original text does not use term “leg” for a side of a right triangle that is not the
hypotenuse. However, we will use this standard English term freely in the translation. –transl.
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