1992 AIME Problems/Problem 14
Contents
Problem
In triangle ,
,
, and
are on the sides
,
, and
, respectively. Given that
,
, and
are concurrent at the point
, and that
, find
.
Solution 1
Let and
Due to triangles
and
having the same base,
Therefore, we have
Thus, we are given
Combining and expanding gives
We desire
Expanding this gives
Solution 2
Using mass points, let the weights of ,
, and
be
,
, and
respectively.
Then, the weights of ,
, and
are
,
, and
respectively.
Thus, ,
, and
.
Therefore:
.
See also
1992 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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