G.120 radius to point of contact is perpendicular to tangent — Heath

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.

For let a straight line / D E / touch the circle ( A B C ) at the point C, let the centre F of the circle ( A B C ) be taken, and let / F C / be joined from F to C; I say that / F C / is perpendicular to / D E /.

For, if not, let / F G / be drawn from F perpendicular to / D E /.

Then, since the angle < F G C ) is right,

[G.42]the angle < F C G ) is acute;

and the greater angle is subtended by the greater side; [G.44]

therefore / F C / is greater than / F G /.

But / F C / is equal to / F B /;

therefore / F B / is also greater than / F G /, the less than the greater: which is impossible.

Therefore / F G / is not perpendicular to / D E /.

Similarly we can prove that neither is any other straight line except / F C /;

therefore / F C / is perpendicular to / D E /.

Therefore etc. Q. E. D.

Dependency Graph

Required for

G.121, G.140, G.141, G.142, G.144, G.145, G.149, G.151, G.159, G.162, G.163, G.164, G.563, G.565, G.570, G.571, G.572, G.573, G.592, G.601