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A line is tangent to two intersecting circles at P and Q. The common chord is extended to meet PQ at T. Prove that T is the midpoint of PQ.
Solution:
To illustrate the problem, it is better to draw the figure as follows
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| Photo by Math Principles in Everyday Life |
Let A and B are the centers of two circles and CD is the common chord of two circles. If CD is extended to meet PQ at T, then CT is the common external segment of the two circles.
If a theorem says "When a secant segment and a tangent segment are drawn to a circle from an external point, the product of the secant segment and its external segment is equal to the square of the tangent segment.", then the working equation for circle A is
and for circle B is
Since CT is the common external segment of two circles, then we can equate the two working equations as follows
Take the square root on both sides of the equation, we have
Since PT ≅ TQ, then T is the midpoint of PQ.