Question

In the figure, two circles touch each other at the point C.Prove that the common tangents to the circles at C, bisect the common tangents at P and Q.

Answer 1

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Answer 2

Given:

• Two circles touch each other at a common point C.

To Find:

• The common tangents to the circle at C bisect the common tangents P and Q.

Solution:

• We can come to few conclusions from the given figure.
• We can see that PR and CR are the common tangents drawn to the same circle externally from point R.
• Therefore, PR = CR → (1)
• Same way, QR and CR are the common tangents drawn to the same circle externally from point R.
• Therefore, QR = CR → (2)
• Equating equations (1) and (2) we get,
• PR = QR
• Here we get to know that R is the mid-point of PQ.
• ∴ The common tangents at C bisect the common tangents P and Q.

Hence Proved

The common tangents at C bisect the common tangents at P and Q.