Euclid's Third Proposition in the Poincaré DiskTo cut off from the greater of two given unequal straight lines a straight line equal to the less.
(Feel free to adjust the length of C and move points A, B, and D)
Let AB and C be the two given unequal straight lines, and let AB be the greater of them.
It is required to cut off from AB the greater a straight line equal to C the less.
Place AD at the point A equal to the straight line C, and describe the circle DEF with center A and radius AD.
Now, since the point A is the center of the circle DEF, therefore AE equals AD.
But C also equals AD, therefore each of the straight lines AE and C equals AD, so that AE also equals C.
Therefore, given the two straight lines AB and C, AE has been cut off from AB the greater equal to C the less.