In the given figure, two circles touch each other at a point C. Prove that the common tangent to the circles at C bisects the common tangent at the points P and Q.
C A B P Q R


Answer:


Step by Step Explanation:
  1. We see that PR and CR are the tangents drawn from an external point R on the circle with center A.
    Thus, PR=CR   (i)

    Also, QR and CR are the tangents drawn from an external point R on the circle with center B.
    Thus, QR=CR   (ii)
  2. From eq (i) and eq (ii), we get
    PR=QR   [Both are equal to CR]

    Therefore, R is the midpoint of PQ.
  3. Thus, we can say that the common tangent to the circles at C bisects the common tangent at the points P and Q.

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