Difference between revisions of "2009 USAMO Problems/Problem 1"
(→Solution 2) |
(→Solution 1) |
| (One intermediate revision by the same user not shown) | |
(No difference)
| |
Latest revision as of 04:31, 27 August 2024
Problem
Given circles 
and 
intersecting at points 
and 
, let 
be a line through the center of intersecting at points 
and 
and let 
be a line through the center of intersecting at points 
and 
. Prove that if 
and lie on a circle then the center of this circle lies on line 
.
Solution 1
Let 
be the circumcircle of 
, 
to be the radius of 
, and 
to be the center of the circle , where 
. Note that 
and 
are the radical axises of 
, 
and 
, respectively. Hence, by power of a point(the power of can be expressed using circle and and the power of can be expressed using circle and ),
![\[O_1O_2^2 - r_2^2 = O_1O_3^2 - r_3^2\]](http://latex.artofproblemsolving.com/4/e/e/4ee0bec89f1a9c5dd34f0c3866c8f3966df4b964.png)
![\[O_2O_1^2 - r_1^2 = O_2O_3^2 - r_3^2\]](http://latex.artofproblemsolving.com/b/9/4/b94c3aed44cbdcbcc6a3e0adc13f60b044c221b7.png)
Subtracting these two equations yields that 
, so must lie on the radical axis of , .
~AopsUser101
See also
| 2009 USAMO (Problems • Resources) | ||
| Preceded by First question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
