Difference between revisions of "Mock AIME 3 Pre 2005 Problems/Problem 14"
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==Solution== | ==Solution== | ||
| + | Invert about a circle with radius 1 and center P. Note that since all relevant circles and lines go through P, they all are transformed into lines, and <math>\omega_1,\omega_2, l</math> are all tangent at infinity (i.e. parallel). That was the crux move; some more basic length chasing using similar triangles gets you the answer. | ||
{{solution}} | {{solution}} | ||
==See also== | ==See also== | ||
Revision as of 23:03, 11 March 2009
Problem
Circles 
and 
are centered on opposite sides of line 
, and are both tangent to at 
. 
passes through , intersecting again at 
. Let 
and 
be the intersections of and , and and respectively. 
and 
are extended past and intersect and at 
and 
respectively. If 
and 
, then the area of triangle 
can be expressed as 
, where 
and 
are positive integers such that 
and are coprime and 
is not divisible by the square of any prime. Determine 
.
Solution
Invert about a circle with radius 1 and center P. Note that since all relevant circles and lines go through P, they all are transformed into lines, and 
are all tangent at infinity (i.e. parallel). That was the crux move; some more basic length chasing using similar triangles gets you the answer.
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