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.
Set where
due to the triangle inequality. By breaking the quadrilateral into
and
, we can create an expression for the area of
. We will use the formula for the area of an equilateral triangle given its side length to find the area of
and Heron's formula to find the area of
.
After simplifying,
Substitute and then the expression becomes
We can ignore the for now and focus on
.
By the Cauchy-Schwarz Inequality,
The RHS simplifies to , meaning the maximum value of
is
.
Finally, the maximum value of the area of is
.
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
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