Difference between revisions of "1977 USAMO Problems/Problem 2"
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== Problem == | == Problem == | ||
| − | <math> ABC</math> and <math> A'B'C'</math> are two triangles in the same plane such that the lines <math> AA',BB',CC'</math> are mutually parallel. Let <math> [ABC]</math> | + | <math> ABC</math> and <math> A'B'C'</math> are two triangles in the same plane such that the lines <math> AA',BB',CC'</math> are mutually parallel. Let <math> [ABC]</math> denote the area of triangle <math> ABC</math> with an appropriate <math> \pm</math> sign, etc.; prove that |
<cmath> 3([ABC]+ [A'B'C']) = [AB'C'] + [BC'A'] + [CA'B']+ [A'BC]+[B'CA] + [C'AB].</cmath> | <cmath> 3([ABC]+ [A'B'C']) = [AB'C'] + [BC'A'] + [CA'B']+ [A'BC]+[B'CA] + [C'AB].</cmath> | ||
| − | == | + | == Hint == |
| − | + | Let the parallel lines <math>AA', BB', CC'</math> be parallel to the <math>x-axis</math>, and choose arbitrary origin. Then we can define <math>A(x_1, a), A'(x_2, a), B(y_1, b), B'(y_2, b), C(z_1, c), C'(z_2, c),</math> 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 == | == See Also == | ||
{{USAMO box|year=1977|num-b=1|num-a=3}} | {{USAMO box|year=1977|num-b=1|num-a=3}} | ||
| + | {{MAA Notice}} | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] | ||
Latest revision as of 21:24, 19 May 2015
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
