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### Geometry Proof Involving Circle and Triangle

```Date: 09/26/2005 at 09:52:33
From: Drew
Subject: Geometry Proofs

Triangle ABC cuts a circle at points E, E', D, D', F amd F'.  Prove
that if AD, BF and CE are concurrent, than AD', BF' and CE' are also
concurrent.

I have researched this and all I come up with is the nine point circle

```

```
Date: 09/28/2005 at 07:55:33
From: Doctor Floor
Subject: Re: Geometry Proofs

Hi, Drew,

Thanks for your question.  We can use two important geometry facts here:

1. Ceva's theorem, see

Ceva's Theorem
http://mathforum.org/library/drmath/view/55095.html

2. The power of a point with respect to a circle:

Intersecting Circles
http://mathforum.org/library/drmath/view/55125.html

Going back to your problem, with Ceva's Theorem we have:

BD   CE   AF
-- * -- * -- = 1 ....................................[1]
DC   EA   FB

While the power of B with respect to the circle yields that:

BD*BD' = BF*BF'

And because FB = -BF and F'B= - BF' this gives us

BD*BD' = FB*F'B

or

BD    F'B
-- =  --- ...........................................[2]
FB    BD'

In the same way we derive that

CE   D'C       AF   E'A
-- = ---  and  -- = --- .............................[3]
DC   CE'       EA   AF'

Now with the help of [2] and [3] we can rewrite [1] into

D'C   E'A   F'B
--- * --- * --- = 1
BD'   CE'   AF'

or equivalently

BD'   CE'   AF'
--- * --- * --- = 1
D'C   E'A   F'B

With (the converse of) Ceva's Theorem this shows that AD', BE' and
CF' are concurrent.

Eric W. Weisstein. "Cyclocevian Triangle."
http://mathworld.wolfram.com/CyclocevianTriangle.html

If you replace "circle" in your question with "conic section" then
the statement is still correct.  The proof is more difficult and
requires some knowledge of "projective geometry".

If you have more questions, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Conic Sections/Circles
College Triangles and Other Polygons
High School Conic Sections/Circles
High School Triangles and Other Polygons

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