2019 AMC 12B Problems/Problem 25
Problem
Let be a convex quadrilateral with and Suppose that the centroids of and form the vertices of an equilateral triangle. What is the maximum possible value of ?
Solution
Set , , as the centroids of , , and respectively, while is the midpoint of line . , , and are collinear due to the centroid. Likewise, , , and are collinear as well. Because and , . From the similar triangle ratios, we can deduce that . The similar triangles implies parallel lines, namely is parallel to .
We can apply the same strategy to the pair of triangles and . We can conclude that is parallel to and . Because , and the pair of parallel lines preserve the 60 degree angle, meaning . Therefore, is equilateral.
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.