Secant, Tangents and Orthogonality:
What Is This About?
A Mathematical Droodle

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Explanation

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Copyright © 1996-2018 Alexander Bogomolny

Secant, Tangents and Orthogonality

The applet suggests the following theorem:

Point P lies outside a circle with center O. Straight lines L1 and L2 intersect at P. Line L1 touches the circle at point A and L2 intersects the circle at points B and C. Tangents to the circle at points B and C intersect at point X. Prove that segments AX and OP are perpendicular.

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Proof 1 (Elementary)

Denote angle AOP by φ, PCO by α, and COP by &beta. Then the law of sines implies

(1) OP/sinα = CO/sinβ = OP·sinφ/sinβ.

Consider the projections of points A and X on line OP - points A' and X' (not shown in the applet.) Calculate the lengths of PA' and PX':

(2) PA' = OP·cos2φ,

whereas

(3)
PX'= OP - OX·sinβ
 = OP - (CO/sinα)·sinβ
 = OP - (CO/sinα)·sinα·sinφ
 = OP - CO·sinφ
 = OP·(1 - sin2φ)
 = OP·cos2φ.

So we see that A' coincides with X'. But the projections of two points on the line OP can coincide if and only if these points lie on the line perpendicular to OP.

Proof 2

We know that A and X are the poles of L1 and L2, respectively. The point of intersection of the two lines (P) lies therefore on the the polars of both A and X. By La Hire's theorem, both A and X lie on the polar of P. But a polar of a point with respect to a circle is perpendicular to the line joining the point with the center of the circle. Q.E.D.

References

  1. D. Fomin, A. Kirichenko, Leningrad Mathematical Olympiads 1987-1991, MathPro Press, 1994, pp. 45, 159.

Poles and Polars

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