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. 1 The 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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