Euclid's Ninth Proposition in the Poincaré Disk
Euclid's Ninth Proposition - http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI9.html 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 AB. Cut off AE from AC equal to AD, and join DE. Construct the equilateral triangle DEF on DE, and join AF. I say that the angle BAC is bisected by the straight line AF. Since AD equals AE, and AF is common, therefore the two sides AD and AF equal the two sides EA and 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.