To bisect a given rectilinear angle.
Let the angle BAC be the given rectilinear angle. It is required to bisect it.
Take an arbitrary point D on line AB. Cut off line AE from line AC equal to line AD, and join line DE. Construct the equilateral triangle DEF on line DE, and join line AF.
I say that the angle BAC is bisected by the straight line AF.
Since line AD equals line AE, and line AF is common, therefore the two sides line AD and line AF equal the two sides line AE and line AF respectively.
And the base DF equals the base EF, therefore the angle DAF equals the angle EAF.
Therefore the given rectilinear angle BAC is bisected by the straight line AF.