Question

Theorem of touching circle
write the proof please ​

Answer 1

Theorem -

If two circles touch each other (internally), then their point of contact lies on the straight line joining their centers.

Note that at the point of contact, a common tangent L can be drawn to both the circles:

Proof -

The justification of this result is quite straightforward :-

AP must be perpendicular to L

and, so must BP

This is because L is the tangent to both the circles at P.

And we now further derive the can following useful result.

Corollary -

For two circles touching each other, the distance between their centres is equal to the sum of their radii (if the circles touch each other externally) or the difference of their radii

(if the circles touch each other internally).

Thank You!

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