In other words, given a rectilineal angle (call it BAC), we must bisect it.

Given:[br]1) Consider line AB,[br]2) and line AC that form the angle BAC.[br][br]Note: We must construct a line AF, such that angle BAC is bisected by the straight line AF.[br][br]3) Let a point D be taken at random on AB.[br]4) Let AE be cut off from AC equal to AD. [I. 3][br]5) Let DE be joined.[br]6) And on DE, let the equilateral triangle DEF be constructed;[br]7) let AF be joined.[br][br]I say that the angle BAC has been bisected by the straight line AF.[br][br]Explanation: For, [br]->since AD is equal to AE,[br]->and AF is common,[br]---->the two sides DA, AF, are equal to the two sides EA, AF respectively.[br][br]And the base DF is equal to the base EF;[br]---->therefore, the angle DAF is equal to the angle EAF. [I. 8][br][br]Therefore, the given rectilineal angle BAC has been bisected by the straight line AF. █