Solution to Monge's Theorem Problem 1
First, construct circle c1 with center O and an arbitrary point Q not on the circle. Next, construct a line through O and Q. Construct a segment OQ and it's midpoint N. Then, construct circle c2 using N as its center and ON as its radius. Construct the points of intersection, P and R, between the circles c1 and c2. Finally, construct segments PQ and QR, which are tangent lines to circle c1 through point Q (labeled as t1 and t2 on the following sketch).
- Tangents to a circle are always perpendicular to the radius of the circle.
- (More of an explanation of Fact 1 than a fact of its own...) Take a line L1 through O, the center of a circle, and form a perpendicular line L2 through Q, a point not on that line. Let point P be the intersection of L1 and L2. By rotating L1, we see that P traces a circle through O and Q. By Fact 1 (stated above), the line containing QP must be tangent to the circle at the points at which the traces of P cross the circle centered at O in only one location, since L1 contains the radius of the circle and is perpendicular to L2.
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