Euclid I-10. (pg 9)

Emily Crum
Explanation: Let AB be the finite straight line. Construct an Equilateral Triangle using AB as the base. (The equilateral triangle was created with the same process as before in Euclid I-1. (pg 3) from last week. Because this has already been proven, it is unnecessary to do so again. ) Continuing on, Triangle ABC is our equilateral triangle. Thus AC = BC. Draw an angle bisector through angle ABC (this was done in the previous construction so once again it is unnecessary to prove again) This bisector creates angles ACE and BCE. Because CE is a bisector, angle ACE = angle BCE. DE is equal to DE. Thus by SAS, Triangle ACE is congruent to Triangle BCE. Because of this congruence, this means that AE = BE. Therefore the finite straight line AB has been bisected.