In Fig. 10.21, two circles touch each other at the point `C` . Prove that the common tangent to the circles at `C` , bisects the common tangent at `P` and `Q` . (FIGURE)

Explanation

In the given figure, `PR` and `CR` are both tangents drawn to the same circle from an external point `R`.
`:. PR=CR`…..`(i)`
Also, `QR` and `CR` are both tangents drawn to the same circle (second circle) from an external point `R`.
`:. QR=CR`........`(ii)`
From `(i)` and `(ii)`, we get
`PR=QR` [each equal to `CR`]
`:. R` is the midpoint of `PQ`.
i.e, the common tangent to the circles at `C`, bisects the common tangent at `P` and `Q` .