Difference between revisions of "2008 Indonesia MO Problems/Problem 1"
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Latest revision as of 17:50, 8 April 2020
Problem
Given triangle . Points outside triangle are chosen such that triangles , , and are equilateral triangles. Prove that cicumcircles of these three triangles are concurrent.
Solution
Let be the intersection of the circumcircles of and . Note that and are cyclic quadrilaterals. Thus, and .
We know that and are equilateral, so . Therefore, , so .
Since is equilateral as well, . Note that , and since the circumcircle is the circle that passes through , point must also be on the same circumcircle of . Thus, the cicumcircles of these three triangles are concurrent.
See Also
2008 Indonesia MO (Problems) | ||
Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 | Followed by Problem 2 |
All Indonesia MO Problems and Solutions |