Proposition 3.17

Proposition 3.17 - From a given point A not on a circle E, draw a line that is tangent to E through the point A

Proof Let A be the given point and circle BCD with center E be the given circle. Join AE, and let the circle with center E and radius AE be drawn. Draw F such that F lies on circle ABC and is collinear with A and E. From F, draw a perpendicular line to AE. The intersection of the line with the circle of radius AE and center E is G. Draw GE and H such that H lies on circle ABC and is collinear with G and E. Draw AH. Now, GE=AE, HE=FE, and <GEA=<AEG. So triangles GEF and AHE are congruent. This means that <GFE, a right angle, is equal to <AHE, another right angle. Therefore, by 3.16, AB is tangent to circle ABC.