Inside of a square ABCD, construct the largest equilateral triangle with one side on AB. Next, fit the largest equilateral triangle in the remaining space with one side on CD.

Find the ratio of the area of the smaller triangle to the largest.

The side of the square is the base 'b' of the large triangle. The height of an equilateral triangle is 1/2 base x sqrt(3), so its area is sqrt(3)/4 b^2. The smaller triangle partially shares a side with the larger. The base of the smaller together with the extended shared side makes a right triangle, which in turn gives us the base of the small triangle as b/sqrt(3). Since area goes as base^2, the ratio of areas is 1/3.

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PS - As per Jer's comment: Thanks Jer for: 1) introducing Thirdsday, 2) Likewise, introducing "Brilliant" (It seems that you contribute to this pay-for site. I wonder how that works.) 3) For the clearly spirited teaching example. Your students are lucky!

*Edited on ***January 12, 2019, 7:12 pm**