# Is it possible to use inversion to solve this USAMO problem in 2007?

I've no previous experience to solve any problems by inversive geometry but I am willing to see how it works. But I think I know some of the basic definition about inversion in geometry. Also I expect to see you to simplify the solution by inversive geometry simply because the solution is the link provide is unreadale and I would like to see if inversion really help to write a concise proof. Thanks in advance! Here is the problem:

Let $ABC$ be an acute triangle with $\omega$, $\Omega$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_A$ is tangent internally to $\Omega$ at $A$ and tangent externally to $\omega$. Circle $\Omega_A$ is tangent internally to $\Omega$ at $A$ and tangent internally to $\omega$. Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and $\Omega_A$, respectively. Define points $P_B$, $Q_B$, $P_C$, $Q_C$ analogously. Prove that

$$8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3$$

with equality if and only if triangle $ABC$ is equilateral triangle. Is there anyone could help me with this, this have asked for 36 hours ago? reference: http://www.artofproblemsolving.com/Wiki/index.php/2007_USAMO_Problems/Problem_6

Is there any easier way to do this since all the USAMO problem i have seen except this one could be finish with just one or two theorems or have very short solution. For example, this can be done without angle chasing but is just a consquence of miquel point thorem:

Could you use Miquel point theorem to continue this solution on a USAMO problem?

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Maybe if you would do a little more than copy/paste, we would be saved [[from]] all the [[excess]] punctuation. –  The Chaz 2.0 Apr 9 '12 at 0:47
@TheChaz - The preview of my post is not working, so i have to post it and then edit it –  Victor Apr 9 '12 at 0:49
And, please spell the words properly--"pervious" should be "previous". And observing rules of Capitalizing helps tremendously. Thank you. –  user21436 Apr 9 '12 at 0:52
Victor, I understand. That can be very frustrating! –  The Chaz 2.0 Apr 9 '12 at 0:55
@Victor I assume you are preparing for USAMO 2012, and since you don't have much time I would suggest you go through this elaborate solution to the problem at web.williams.edu/go/math/sjmiller/public_html/greenchicken/…. This reference not only contains solution to your question (page 28-29), but it also has some good set of problems/solutions. –  Kirthi Raman Apr 12 '12 at 14:21