Difference between revisions of "2012 AIME II Problems/Problem 15"
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== Solution == | == Solution == | ||
− | Use angle bisector theorem to find <math>CD=21/8</math>, <math>BD=35/8</math>, and <math>AD=15/8</math>. Use Power of the Point to find <math> | + | Use angle bisector theorem to find <math>CD=21/8</math>, <math>BD=35/8</math>, and <math>AD=15/8</math>. Use Power of the Point to find <math>DE=49/8</math>, and so <math>AE=8</math>. Use law of cosines to find <math>\angle CAD = \pi /3</math>, hence <math>\angle BAD = \pi /3</math> as well, and <math>\triangle BCE</math> is equilateral. |
I'm sure there is a more elegant solution from here, but instead we do'll some hairy law of cosines: | I'm sure there is a more elegant solution from here, but instead we do'll some hairy law of cosines: |
Revision as of 00:00, 18 April 2012
Problem 15
Triangle is inscribed in circle
with
,
, and
. The bisector of angle
meets side
at
and circle
at a second point
. Let
be the circle with diameter
. Circles
and
meet at
and a second point
. Then
, where
and
are relatively prime positive integers. Find
.
Solution
Use angle bisector theorem to find ,
, and
. Use Power of the Point to find
, and so
. Use law of cosines to find
, hence
as well, and
is equilateral.
I'm sure there is a more elegant solution from here, but instead we do'll some hairy law of cosines:
(1)
Adding these two and simplifying we get:
(2). Ah, but
(since F lies on the same circle), and we can find
using the law of cosines:
, and plugging in
we get
.
Also, , and
(since
is on the circle with diameter
), so
.
Plugging in all our values into equation (2), we get:
, or
.
Finally, we plug this into equation (1), yielding:
Thus,
or
The answer is
.
See Also
2012 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |