1992 AIME Problems/Problem 14
Contents
[hide]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 \begin{align*}
\frac{AO}{OA'}=\frac{K_B+K_C}{K_A}\\ \frac{BO}{OB'}=\frac{K_A+K_C}{K_B}\\ \frac{CO}{OC'}=\frac{K_A+K_B}{K_C}.
\end{align*} Thus, we are given Combining and expanding gives
We desire
Expanding 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.