Difference between revisions of "1989 AIME Problems/Problem 10"
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Revision as of 20:24, 11 November 2007
Problem
Let ,
,
be the three sides of a triangle, and let
,
,
, be the angles opposite them. If
, find
![$\frac{\cot \gamma}{\cot \alpha+\cot \beta}$](http://latex.artofproblemsolving.com/8/5/8/858c49b569dc76c503dbdbfa8d0b78fd2c04fe1d.png)
Solution
We can draw the altitude h to c, to get two right triangles.
, from the definition of the cotangent.
From the definition of area, , so therefore
Now we evaluate the numerator:
.
, from the Law of Cosines
, where R is the circumradius.
Since ,
See also
1989 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 9 |
Followed by Problem 11 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |