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.