III.18 Proposition III.18
If a straight line touch a circle, and a straight line be joined from the centre to the point of contact, the straight line so joined will be perpendicular to the tangent.
Proof
Let line AB touch the circle at C, with centre F. Suppose FC
is not perpendicular to AB; drop the perpendicular FG to AB
at some point G=C. In right triangle △FGC
(right-angled at G), FC is the hypotenuse, so by I.19, FG<FC. But G lies on the tangent AB, which has no point inside
the circle (Definition III.2); so FG≥FC= radius. The two
inequalities contradict. Hence G=C and FC⊥AB.
lines 74–74 in main.tex
