AIME I Problem 9 2007

[zefuri]

n right triangle ABC with right angle C, CA=30 and CB=16. Its legs [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are extended beyond A and B. Points [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center [unparseable or potentially dangerous latex formula] is tangent to the hypotenuse and to the extension of leg CA, the circle with center [unparseable or potentially dangerous latex formula] is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as p/q, where p and q are relatively prime positive integers. Find p+q.

[AuSmith]

I was trying to draw the description in this problem and the tangencies were always strained. It kept coming out like this.

http://s151.photobucket.com/albums/s126 ... titled.gif

Then, I said, "Heck, I'm going to try a different way." And got something that looks like a hoot owl. Here's how it looked (no embellishments).

http://s151.photobucket.com/albums/s126 ... itled2.gif

Well, it turns out the hoot owl version is the one they wanted described.




Heh, well, if you want something significantly different than the AoPS solution, and a more serious picture,

http://s151.photobucket.com/albums/s126 ... itled3.gif

The total length of the green line is the same as the hypotenuse (which conveniently evaluates to 34 (from the 8,15,17 triangle)). We can find the measures of the blue angles easily, but really we want their tangents so we can get the lengths of the other two green segments in terms of radius R. We'll call the left blue angle [unparseable or potentially dangerous latex formula] and the right blue angle [unparseable or potentially dangerous latex formula], while [unparseable or potentially dangerous latex formula] are what they usually are in triangle-speak.

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]

We want tangents of those buggers, so it would be good to find a half angle formula for tangent. I like to derive it.
=================================================
[unparseable or potentially dangerous latex formula]

and

[unparseable or potentially dangerous latex formula]

give you

[unparseable or potentially dangerous latex formula]

and

[unparseable or potentially dangerous latex formula]

and

[unparseable or potentially dangerous latex formula]
=================================================

Now, back to our blue angles.

[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]

The total length of the green segments becomes

[unparseable or potentially dangerous latex formula]

[unparseable or potentially dangerous latex formula]