Euclid I-12. (pg 10)

Emily Crum
Explanation: Draw line AB. Create pt C not on the line AB. Create pt. D at random on the other side of the AB line. Create a circle with C as the center and D as a point on the circumference. Label where the circle intersects the line AB as E and F. Connect EC and CB. Bisect EF with point G and connect CG. Since EC and CB are radii of the circle, they are equal. EG=GF because G bisects EF. CG=CG Thus by SSS, triangle ECG = triangle BCG. Because of this congruency, angle EGC=BCG. It was proven earlier (Proposition 11) that these 2 angles are right angles. Therefore because EGC and BCG are right angles, by definition or perpendicular, CG is perpendicular to AB.