# Thread: Butterfly Theorem

1. ## Butterfly Theorem

For my college geometry class, I have to do a write up of a proof of the Butterfly Theorem. I'm doing a write up of the first proof given on that Page. It's pretty easy, but there's one thing that's bothering me. I know that inscribed angles subtended by the same chord are equal, but I can't find any theorem anywhere that says this. It isn't something that has been specifically stated in our class, and I'd rather say that this is true because of a theorem instead just stating that fact and not backing it up.

Does anyone here know what theorem this is, or any theorem that exists that has this fact as a corollary?

I found something that came from Euclid's Elements that says, "In a circle the angles in the same segment equal one another." Is that basically what I'm looking for, but with just different wording?

Also, I'm not quite clear on how they justify this step:
AX·XD/ CY·YB = PX·XQ/ PY·YQ

Thanks for all your help.

2. You have said that this is a College Geometry class.
If your instructor is anything like I am saying “inscribed angles subtended by the same chord are equal” is just a no-no. Inscribed angles subtended by the same chord are congruent. This is easily proven: observe that we have two isosceles triangles that must be congruent.

3. Thanks, I'll make sure to change this in my write up. We haven't really discussed much about triangles, really, in this class. We've basically just done alot of affine transformations, so I doubt she would have cared too much, but this will probably sound better.

Any insight on why this equation is true?
AX·XD/ CY·YB = PX·XQ/ PY·YQ

Also, now that I look closer at the proof, how is this step justified?
(a^2 - x^2)/(a^2 - y^2) = a^2/a^2

edit: If anyone thinks they can help me with this and wouldn't mind doing this over AIM, my screen name is greek master 500.

4. Yeah, I've seen that proof. Unfortunately, I have to do a writeup of the proof I linked in my original post. Thanks anyways though.

5. Ok, I figured out the AX·XD/ CY·YB = PX·XQ/ PY·YQ part. It has to do with the power of a point theorem.

I however still do not know how to justify the "(a^2 - x^2)/(a^2 - y^2) = a^2/a^2" part.

6. Eh, I figured it out. Thanks to everyone who spent time on this!

7. Originally Posted by galactus
I already used that site as a link

This is my 32th Post!!!