2019 AIME II Problems/Problem 15
Problem
In acute triangle points
and
are the feet of the perpendiculars from
to
and from
to
, respectively. Line
intersects the circumcircle of
in two distinct points,
and
. Suppose
,
, and
. The value of
can be written in the form
where
and
are positive integers, and
is not divisible by the square of any prime. Find
.
Solution
First we have , and
Similarly,
and dividing these each by
gives
.
It is known that the sides of the orthic triangle are , and its angles are
,
, and
. We thus have the three sides of the orthic triangle now.
Letting
be the foot of the altitude from
, we have, in
,
similarly, we get
To finish,
The requested sum is .
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Solution 1
Let
Therefore
By power of point, we have
Which are simplified to
Or
(1)
Or
Let
Then,
In triangle , by law of cosine
Pluging (1)
Or
Substitute everything by
The quadratic term is cancelled out after simplified
Which gives
Plug back in,
Then
So the final answer is
By SpecialBeing2017
Solution 2
Let and
By power of point, we have
and
Therefore, substituting in the values:
Notice that quadrilateral is cyclic.
From this fact, we can deduce that and
Therefore is similar to
.
Therefore:
Now using Law of Cosines on we get:
Notice
Substituting and Simplifying:
Now we solve for using regular algebra which actually turns out to be very easy.
We get and from the above relations between the variables we quickly determine
,
and
Therefore
So the answer is
By asr41
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 14 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
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