Problem 29
Two circles touch internally at C. Chords CP and CQ of the larger circle are drawn, intersecting the smaller circle at A and B respectively. If PQ is a tangent to the smaller circle at R, show that:
AC/BC = PR/QR
Here’s a simulation that illustrates the problem.
Try dragging one of the blue points. Is AC/BC always equal to PR/QR? How can we prove the same?
Solution
Here are some tools that can help you solve the problem.
Basic Proportionality Theorem
In a triangle ABC, if points D and E are taken on AB and AC respectively such that DE || BC, then:
AD/BD = AE/CE
Here’s a simulation that demonstrates the theorem.
Here, DE is parallel to BC. Try dragging the vertices A, B, and C, or the segment DE (using the blue point in the middle). Do the ratios AD/BD and AE/CE always remain equal?
Alternate Segment Theorem
If a tangent is drawn to a circle at the end point of its chord PQ, then the angle between the tangent and PQ is the same as PQ subtends in the alternate segment of the circle.
Too much information packed into once sentence? Have a look at the simulation to understand this theorem.
You can drag the points P, Q, and R. Is the angle between PQ and the tangent same as the angle subtended in the alternate segment?
Tangent Secant Theorem
Let P be a point outside a circle. Let PT be a tangent to the circle and PAB be a secant, where A and B lie on the circle. Then,
PT² = PA.PB
Here’s a simulation that demonstrates the theorem.
Try dragging the points P and A. Does PT² always remain equal to PA.PB?
Now, to proceed, join A and B (in the original figure). Can you apply the above three theorems to prove the result?
Were you able to solve it using the above hints? Or, did you solve using any other method? Or, do you need some more help? You can reach out at the Telegram group.
Problem Source: Math Club
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