1988 AIME Problems/Problem 12
Contents
Problem
Let be an interior point of triangle and extend lines from the vertices through to the opposite sides. Let , , , and denote the lengths of the segments indicated in the figure. Find the product if and .
Solution 1
Call the cevians AD, BE, and CF. Using area ratios ( and have the same base), we have:
Similarily, and .
Then,
The identity is a form of Ceva's Theorem.
Plugging in , we get
Solution 2
Let the labels be the weights of the vertices. First off, replace with . We see that the weights of the feet of the cevians are . By mass points, we have that:
If we add the equations together, we get
If we multiply them together, we get
Multiplying both sides by , we get that .
See also
1988 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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