Proofs of Euclid I. 5

Abstract

SEVERAL writers have lately expressed their opinions in favour of replacing the present proof of this proposition by an alternative proof based on the supposition that the bisector of the vertical angle of the isosceles triangle is drawn, irrespective of the fact that no construction has been given for drawing this bisector. Now there may be some advantage in using a “hypothetical construction” to prove a proposition, where its avoidance necessitates a long and tedious alternative proof. In the present instance the artifice is absolutely unnecessary, as the proof can be simplified in any of the following ways, A being the vertical angle of the isosceles triangle ABC:—