The Mohr-Mascheroni Theorem (and also here) states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone.
A key point in the proof of this theorem is the bisection of a given arc with compass alone.
How to achieve this is explained here.
The proof heavily depends on the Pythagorean Theorem.
I am interested in a proof, if any, of this key point which avoids the Pythagorean Theorem.