1977 USAMO Problems/Problem 2
Problem
and 
are two triangles in the same plane such that the lines 
are mutually parallel. Let ![$[ABC]$](http://latex.artofproblemsolving.com/d/3/3/d33cc80fa8f093e155c5be46d2e5d9da3d7e1ef5.png)
denote the area of triangle with an appropriate 
sign, etc.; prove that
![\[3([ABC]+ [A'B'C']) = [AB'C'] + [BC'A'] + [CA'B']+ [A'BC]+[B'CA] + [C'AB].\]](http://latex.artofproblemsolving.com/5/8/b/58b0b0c6ba5c2cb0d7264799fd86e5385a1e5919.png)
Hint
Let the parallel lines 
be parallel to the 
, and choose arbitrary origin. Then we can define 
and so, by the area of a triangle formula, it suffices to prove an algebraic statement that is readily shown to be true.
See Also
| 1977 USAMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: Unable to save thumbnail to destination
