2019 AMC 12B Problems/Problem 25

Problem

Let $ABCD$ be a convex quadrilateral with $BC=2$ and $CD=6.$ Suppose that the centroids of $\triangle ABC,\triangle BCD,$ and $\triangle ACD$ form the vertices of an equilateral triangle. What is the maximum possible value of $ABCD$ ?

$\textbf{(A) } 27 \qquad\textbf{(B) } 16\sqrt3 \qquad\textbf{(C) } 12+10\sqrt3 \qquad\textbf{(D) } 9+12\sqrt3 \qquad\textbf{(E) } 30$

Solution

Set $G_1$ , $G_2$ , $G_3$ as the centroids of $ABC$ , $BCD$ , and $CDA$ respectively, while $M$ is the midpoint of line $BC$ . $A$ , $G_1$ , and $M$ are collinear due to the centroid. Likewise, $D$ , $G_2$ , and $M$ are collinear as well. Because $AG_1 = 3AM$ and $DG_2 = 3DM$ , $\triangle MG_1G_2\sim\triangle MAD$ . From the similar triangle ratios, we can deduce that $AD = 3G_1G_2$ . The similar triangles implies parallel lines, namely $AD$ is parallel to $G_1G_2$ .

We can apply the same strategy to the pair of triangles $\triangle BCD$ and $\triangle ACD$ . We can conclude that $AB$ is parallel to $G_2G_3$ and $AB = 3G_2G_3$ . Because $3G_1G_2 = 3G_2G_3$ , $AB = AD$ and the pair of parallel lines preserve the 60 degree angle, meaning $\angle BAD = 60^\circ$ . Therefore, $\triangle BAD$ is equilateral.

Set $BD = 2x$ where $2\leq x\leq 4$ due to the triangle inequality. By breaking the quadrilateral into $[ABD]$ and $[BCD]$ , we can create an expression for the area of $[ABCD]$ . We will use the formula for the area of an equilateral triangle given its side length to find the area of $[ABD]$ and Heron's formula to find the area of $[BCD]$ .

After simplifying,

$[ABCD] = x^2\sqrt 3 + \sqrt{36 - (x^2-10)^2}$

Substitute $k = x^2 - 10$ and then the expression becomes

$[ABCD] = k\sqrt{3} + \sqrt{36 - k^2} + 10\sqrt{3}$

We can ignore the $10\sqrt{3}$ for now and focus on $k\sqrt{3} + \sqrt{36 - k^2}$ .

By the Cauchy-Schwarz Inequality,

$(k\sqrt 3 + \sqrt{36-k^2})^2 \leq ((\sqrt{3})^2+1^2)((k)^2 + (\sqrt{36-k^2})^2).$

The RHS simplifies to $12^2$ , meaning the maximum value of $k\sqrt{3} + \sqrt{36 - k^2}$ is $12$ .

Finally, the maximum value of the area of $[ABCD]$ is $\boxed{\textbf{(C) }12 + 10\sqrt{3}}$ .

2019 AMC 12B (Problems • Answer Key • Resources)
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2019 AMC 12B Problems/Problem 25

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