Mock AIME 3 Pre 2005 Problems/Problem 11
Contents
[hide]Problem
is an acute triangle with perimeter
.
is a point on
. The circumcircles of triangles
and
intersect
and
at
and
respectively such that
and
. If
, then the value of
can be expressed as
, where
and
are relatively prime positive integers. Compute
.
Solution
Remark that since is cyclic we have
, and similarly
. Therefore by AA similarity
. Thus there exists a spiral similarity sending
to
and
to
, so by a fundamental theorem of spiral similarity
. The angle equality condition gives
, so
is isosceles and
. Similarly,
. Finally, note that the congruent side lengths actually imply
, so
.
Let and
. Remark that from the perimeter condition
. Now from Power of a Point we have the system of two equations
Expanding the second equation and rearranging variables gives
. Back-substitution yields
and consequently
. Thus
and
, so the desired ratio is
.
Solution 2
$\documentclass{article} \usepackage{fourier} \begin{document}$ (Error compiling LaTeX. Unknown error_msg)
Notice that $\angle{BCF} = \angle{DAF} = \frac{\widearc{DF}}{2}$ (Error compiling LaTeX. Unknown error_msg) and $\angle{EBC} = \angle{EBD} = \angle{EAD} =\frac{\widearc{DE}}{2}$ (Error compiling LaTeX. Unknown error_msg). Hence, . Furthermore, through cyclic quadrilaterals, we can find that
and
.
$\end{document}$ (Error compiling LaTeX. Unknown error_msg)
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
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