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^@.
Answer:
- We see that ^@PR^@ and ^@CR^@ are the tangents drawn from an external point ^@R^@ on the circle with center ^@A^@.
Thus, ^@ PR = CR \space \space \space \ldots \text{(i)}^@
Also, ^@QR^@ and ^@CR^@ are the tangents drawn from an external point ^@R^@ on the circle with center ^@B^@.
Thus, ^@ QR = CR \space \space \space \ldots \text{(ii)} ^@ - From ^@ eq \space \text{(i)} ^@ and ^@ eq \space \text{(ii)} ^@, we get
^@ PR = QR \space \space \text{ [Both are equal to CR]} ^@
Therefore, ^@R^@ is the midpoint of ^@PQ^@. - Thus, we can say that the common tangent to the circles at ^@C^@ bisects the common tangent at the points ^@P^@ and ^@Q^@.
