AMC 10 · 2019 · #25
Grade 11 geometry-2dPick an answer.
The question asks for a maximum over a whole family of shapes, so Tool #14 (Extreme Principle) is the destination: get the area down to one expression in one variable, then push that variable to its best value and check the best value is legal. Getting there needs the other tools first. Tool #1 (Draw a Diagram) exposes the fact that all three named triangles contain the vertex C, so the three centroids can be measured from C. Tool #15 (Organize Information in More Ways) re-reads those three centroids as a shrunken copy of the midpoint triangle of △ ABD, which converts the strange equilateral condition into the clean statement "△ ABD is equilateral". Tool #7 (Identify Subproblems) cuts ABCD along the diagonal BD into two triangles whose areas can be handled separately. Tool #4 (Introduce a Variable) names the one remaining degree of freedom, the angle at C. Tool #13 (Convert to Algebra) turns both areas into functions of that angle and collapses the sum into a single sine wave, whose maximum is then read off.
Find the shared vertex
All three contain one point.
Three triangles sharing one vertex give three medians from that same vertex, so all three centroids are pinned to rays leaving C.
10.G-CO.C.10Draw A DiagramDilate from that point
Dilating carries the equilateral condition back.
Blowing the centroid triangle up from C lands exactly on the midpoint triangle, and a midpoint triangle is just the original shrunk by half.
Blowing the centroid triangle up from the shared vertex lands it exactly on the midpoint triangle.
▸ Why?
Scaling from one point stretches every length by the same factor, so the shape is untouched.
▸ Why?
The enlarged triangle has the same angles as the original, so it is a genuine scaled copy.
Cut along the diagonal
Split the quadrilateral into two triangles.
A diagonal turns an awkward quadrilateral into two triangles that share one length, and that shared length is the only bridge between them.
6.G.A.1Identify SubproblemsName the angle
Name the angle between the fixed sides.
One angle controls the whole picture, and the Law of Cosines hands back k² — precisely the quantity the equilateral area formula wants.
11.G-SRT.D.11Introduce A VariableAdd the second triangle
The area becomes a function of that angle.
The same angle drives both pieces — a sine for the triangle it opens and a cosine for the diagonal it stretches.
11.G-SRT.D.9Convert To AlgebraCollapse into one sine wave
The two terms fold into one sine wave.
A sine plus a cosine of the same angle is still just one wave, only shifted, so its size is capped by its amplitude.
11.F-TF.C.9Convert To AlgebraPush it to the maximum
The maximum is twelve plus ten root three.
Once the area is one sine wave, the maximum is free — the only real work left is proving the winning angle builds a legal figure.
9.F-IF.B.4Extreme PrincipleThe equilateral centroid triangle is just a shrunken midpoint triangle, so it secretly says ABD is equilateral; after that the whole area is one sine wave in the angle at C, and its peak is 12 + 10√(3).
- Every named triangle contains C
- Scale up from C by 3/2
- Cut along the diagonal BD
- Name the angle at C
- Add the second triangle
- Collapse into one sine wave
- Push the angle to the extreme