1995 AIME Problems/Problem 4
From AoPSWiki
Problem
Circles of radius
and
are externally tangent to each other and are internally tangent to a circle of radius
. The circle of radius
has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.
![Click to view code [Asy_image]](http://alt1.artofproblemsolving.com/Forum/latexrender/pictures/3/6/5/365615078912c278aba530ed33f1a0d45d49821b.png)
Solution
We label the points as following: the centers of the circles of radii
are
respectively, and the endpoints of the chord are
. Let
be the feet of the perpendiculars from
to
(so
are the points of tangency). Then we note that
, and
. Thus,
(consider similar triangles). Applying the Pythagorean Theorem to
, we find that
![Click to view code [Asy_image]](http://alt2.artofproblemsolving.com/Forum/latexrender/pictures/a/1/4/a14a9884bc09c6a32c30d71a84d2a61343c6ff7d.png)
See also
| 1995 AIME (Problems • Resources) | ||
| Preceded by Problem 3 | Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||




