This Proposition may be proved as follows:
At every point on a circle the tangent is perpendicular to the radius.
Let P and Q be two consecutive points on the circumference. Join CP, CQ, PQ; produce PQ both ways. Now since P and Q are consecutive points, PQ is a tangent (Def. iii.). Again, the sum of the three angles of the triangle CPQ is equal to two right angles; but the angle C is infinitely small, and the others are equal. Hence each of them is a right angle. Therefore the tangent is perpendicular to the diameter.
Or thus: A tangent is a limiting position of a secant, namely, when the secant moves out until the two points of intersection with the circle become consecutive; but the line through the centre which bisects the part of the secant within the circle [iii.] is perpendicular to it. Hence, in the limit the tangent is perpendicular to the line from the centre to the point of contact.
Or again: The angle CPR is always equal to CQS; hence, when P and Q come together each is a right angle, and the tangent is perpendicular to the radius.
Exercises.
1. If two circles be concentric, all chords of the greater which touch the lesser are equal.
2. Draw a parallel to a given line to touch a given circle.
3. Draw a perpendicular to a given line to touch a given circle.