| If a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent. | ||
| For let a straight line DE touch the circle ABC at the point C. Take the center F of the circle ABC, and join FC from F to C. | III.1 | |
| I say that FC is perpendicular to DE. | ||
| For, if not, draw FG from F perpendicular to DE. | I.12 | |
| Then, since the angle FGC is right, the angle FCG is acute, and the side opposite the greater angle is greater, therefore FC is greater than FG. | I.17
I.19 | |
| But FC equals FB, therefore FB is also greater than FG, the less greater than the greater, which is impossible.
Therefore FG is not perpendicular to DE. | ||
| Similarly we can prove that neither is any other straight line except FC. Therefore FC is perpendicular to DE. | ||
| Therefore if a straight line touches a circle, and a straight line is joined from the center to the point of contact, the straight line so joined will be perpendicular to the tangent. | ||
| Q.E.D. | ||
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