Prove that if two circles touch one another, then these chords, drawn from the tangency point, are proportional. Prove that if two circles touch one another, then chords of the internal circle, drawn from the tangency point, are proportional to the chords of the outer circle that you get when you extend the chords of the internal circle.
I hope I've made the question clear.
Thanks.


The ratio of the chords is equal to the ratio of the diameters.

(Thus, the ratio depends only on the diameters, not the specific position of the chords.)
Another proof without words, incorporating @abiessu's observation that the circles need not be internally tangent.

