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Tangents and cords problem

Could someone give me some hints to prove that the two cords AB and CD are congruent. As shown on the picture the two circles can be any relative diameter or distance and the cords from the intersection points of the tangents are always congruent.

Thanks.

Re: Tangents and cords problem

Quote:

Originally Posted by

**BERMES39** Could someone give me some hints to prove that the two cords AB and CD are congruent. As shown on the picture the two circles can be any relative diameter or distance and the cords from the intersection points of the tangents are always congruent.

Thanks.

(Thinking out loud...)

Is there a way you could show that AD and BC are congruent? If so, that means that you would have a rectangle...

1 Attachment(s)

Re: Tangents and cords problem

Quote:

Originally Posted by

**BERMES39** Could someone give me some hints to prove that the two cords AB and CD are congruent. As shown on the picture the two circles can be any relative diameter or distance and the cords from the intersection points of the tangents are always congruent.

Thanks.

1. I've attached a sketch and for convenience I'm refering to the labels of my sketch.

2. If x = y then $\displaystyle \frac xy = 1$

3. You are dealing with 2 right triangles with the same hypotenuse, that means

$\displaystyle d^2 = t^2+r^2~\wedge~d^2=R^2+T^2$

4. In each right triangle you find 2 similar triangles. Use proportions:

$\displaystyle \frac xR = \frac rd~\wedge~\frac yr = \frac Rd$

$\displaystyle x = \frac{ rR}d~\wedge~y = \frac{r R}d$

So $\displaystyle \frac xy=\frac dd=1$