A simple "Geometry Revisited" problem...

[Labib]

After Munir Sir's life threatening speeches(!) @ the special camp, had to visit the BDMO forum....
I've started to solve "Geometry Revisited" exercises,so I'm gonna try to post problems from there...both problems that I liked and problems I can't solve.

This is from the very first exercise of the book....
I really liked it and it seemed a pretty good problem to start with...

$PROBLEM :$

Let $p$ and $q$ be the radii of two circles through $A$, touching $BC$ at $B$ and $C$, respectively.
Then prove that, $pq=R^2$.

Here, $R$ is the circumcircle of the $\triangle ABC$.

[*Mahi*]

Hey , please state the problem clearly. Here you missed to mention what $R$ is.

[Labib]

Sorry about that. It was a mistake.

I will edit it right away!!

[Labib]

BTW, Mahi, I've solved it. I used the law of sines.(It was a problem relating the law of sines).
So you might consider posting your full proof. You can always hide it as you did with the hint.

[*Mahi*]

BTW, Mahi, I've solved it. I used the law of sines.(It was a problem relating the law of sines).
So you might consider posting your full proof. You can always hide it as you did with the hint.

The hint was for other people... I don't think you took that personally... though I've deleted it.

[Labib]

Hey, why would I take it personally bro....
I do like using hints...

[*Mahi*]

You wrote it wrong again
$R$ would be the circumradius.

[Labib]

Sorry, I don't have the editing option now.... :-s

[photon]

i have got a proof,but i want to confirm...is $BC$ a tangent?mine is for tangent $BC$.
@labib bhai,if i write my real name ,then ''photon'' may lose its mystery...isn't $''photon''$ good,huh?

[Labib]

Photon ain't bad, But I ain't Sherlock Homes. I can't stand mysteries.
So you better tell me your name in PM.
(I guess you might one of those two whizzes from Khulna, though I might not be correct.)
And YES, $BC$ is a tangent of the circles with those two radii.