Date: Sun, 20 Oct 1996 12:49:53 +0800
Sender: Rita 
Subject: Secondary Level Math. Problems (Student)
I wonder if there is any proof for this theorem - A tangent to a circle is perpendicular to the radius at the point of contact. If there is any proof for that, can you tell me please?
Hi again Rita:

We can show you two proofs.

An intuitive proof is based on the idea that a circle is symmetric about any diameter: Let O be the center of circle C, and let T on C be the point of tangency. Then if OT is taken to be a mirror, reflection in that mirror will take the tangent into itself (since there is only one line tangent to C at T). This forces the tangent to be perpendicular to the mirror.

Euclid gives an argument in proposition 18 of Book III. Earlier he proved that if a line and a circle intersect then they do so in either one point or two points. If a line intersects a circle in exactly one point T, he defines this line to be the tangent to the circle at the point T. Since this line intersects the circle only once any point on the line other than T is outside the circle. Thus if G is a point on this tangent line different from T then the distance OG is graeter than the distance OT. But an earlier theorem [I. 17] says that of all the distances OG from O to a point G on a line, the shortest is when G is the foot of the perpendicular from O to that line. Thus OT is perpendicular to the tangent line at T.

Chris Fisher and Harley Weston

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