The Elements

...of the three angles, the two are greater than the third; but this is absurd. Therefore, the angles bag and dez are not unequal; they are therefore equal. If, therefore, two triangles have two sides equal to two sides, each to each, and have the base equal to the base, they will also have the angle equal to the angle contained by the equal straight lines; which was to be demonstrated.

Two adjacent geometric triangles of equal size, illustrating the congruent triangles of Euclid's Proposition 8.

9

To bisect a given rectilinear angle. Let the given rectilinear angle be bag; it is required, then, to bisect it. Let an arbitrary point d be taken on ab, and from ag let ae be cut off equal to ad, and let de be joined, and on de let an equilateral triangle dez be constructed, and let az be joined; I say that the angle bag has been bisected by the straight line az. For since ad is equal to ae, and az is common, the two sides da and az are equal to the two sides ea and az, each to each; and the base dz is equal to the base ez; therefore, the angle daz is equal to the angle eaz. Therefore, the given rectilinear angle bag has been bisected by the straight line az; which was to be done.

Library Stamp: A circular faded institutional seal is visible to the right of the diagram for Proposition 9.

A geometric diagram for Proposition 9 showing the bisection of an angle. An angle at vertex A is formed by lines AB and AC. A smaller triangle DEF is constructed within, and a bisecting line AZ runs from the vertex A through the center of the construction.

10

To bisect a given finite straight line. Let the given finite straight line be ab; it is required, then, to bisect the straight line ab. Let an equilateral triangle abg be constructed on it, and let the angle agb be bisected by the straight line gd; I say that the straight line ab is bisected at the point d. For since ag is equal to gb, and gd is common, the two sides ag and gd are equal to the two sides bg and gd, each to each; and the angle agd is equal to the angle bgd; therefore, the base ad is equal to the base bd. Therefore, the given finite straight line ab has been bisected at the point d; which was to be done.

A geometric diagram for Proposition 10 showing the bisection of a straight line. An equilateral triangle ABC is constructed on the base line segment AB. A perpendicular line segment CD bisects the triangle and the base AB at point D.