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,
(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.)
|What if applet does not run?|
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,
Copyright © 1996-2017 Alexander Bogomolny