Euclid's Elements - Book 1 - Proposition 9
To bisect a given rectilineal angle.
In other words, given a rectilineal angle (call it BAC), we must bisect it.
Steps of the construction
Given:
1) Consider line AB,
2) and line AC that form the angle BAC.
Note: We must construct a line AF, such that angle BAC is bisected by the straight line AF.
3) Let a point D be taken at random on AB.
4) Let AE be cut off from AC equal to AD. [I. 3]
5) Let DE be joined.
6) And on DE, let the equilateral triangle DEF be constructed;
7) let AF be joined.
I say that the angle BAC has been bisected by the straight line AF.
Explanation: For,
->since AD is equal to AE,
->and AF is common,
---->the two sides DA, AF, are equal to the two sides EA, AF respectively.
And the base DF is equal to the base EF;
---->therefore, the angle DAF is equal to the angle EAF. [I. 8]
Therefore, the given rectilineal angle BAC has been bisected by the straight line AF. █