Secant, Normal, Tangent

The applet below illustrates the configuration of a circle C(O), with center O), and a secant through points P and Q on the circle. From another point S on the circle a perpendicular SN is droped to PQ. There are two locations of S where SN is perpendicular (normal) to PQ. In both cases, OS||PQ. Assuming S is near to P than to Q, ∠PQS = ∠SQT, where QT is a diameter of the circle; otherwise, ∠QPS = ∠SPT.

(In the applet, the circle can be modified by dragging either P or Q. Point S plays a double role. When the box "Adjust circle" is checked, moving S redefines the circle. If, the box "Move S on circle" is checked, the circle is fixed and S is constrained to stay on the circle.)

When SN is tangent to the circle at S, the radius OS is normal to SN which, in turn, is perpendicular to PQ by the construction. Therefore OS||PQ.

Now assume S is near P than Q. ΔSOQ is isosceles, ∠OSQ = ∠OQS. Angles OSQ and PQS being alternate for the parallels OS and PQ and the transversal SQ are also equal.

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