Difference between revisions of "2021 USAJMO Problems/Problem 2"
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− | We first claim that the three circles <math>(BCC_1B_2),</math> <math>(CAA_1C_2),</math> and <math>(ABB_1A_2)</math> | + | We first claim that the three circles <math>(BCC_1B_2),</math> <math>(CAA_1C_2),</math> and <math>(ABB_1A_2)</math> share a common intersection. |
Let the second intersection of <math>(BCC_1B_2)</math> and <math>(CAA_1C_2)</math> be <math>K</math>. Then | Let the second intersection of <math>(BCC_1B_2)</math> and <math>(CAA_1C_2)</math> be <math>K</math>. Then |
Revision as of 22:00, 18 April 2021
Problem
Rectangles and are erected outside an acute triangle Suppose thatProve that lines and are concurrent.
Solution
We first claim that the three circles and share a common intersection.
Let the second intersection of and be . Then which implies that is cyclic as desired.
Now we show that is the intersection of and Note that so are collinear. Similarly, and are collinear, so the three lines concur and we are done.
~Leonard_my_dude
See Also
2021 USAJMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.