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\begin{document}

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\section*{three squares and two equilateral triangles solution}

\hypertarget{solution_to_the_three_squares_and_two_equilateral_triangles_puzzle}{}\section*{{Solution to the [[Three Squares and Two Equilateral Triangles]] Puzzle}}\label{solution_to_the_three_squares_and_two_equilateral_triangles_puzzle}

[[ThreeSquaresandTwoEquilateralTriangles.png:pic]]

\begin{quote}%
$3$ squares, $2$ equilateral triangles. What's the pink area?


\end{quote}
\hypertarget{solution_by_areas_of_similar_shapes_area_dissection_and_properties_of_equilateral_triangles}{}\subsection*{{Solution by Areas of [[Similar Shapes]], Area [[Dissection]] and Properties of [[Equilateral Triangles]]}}\label{solution_by_areas_of_similar_shapes_area_dissection_and_properties_of_equilateral_triangles}

[[ThreeSquaresandTwoEquilateralTrianglesLabelled.png:pic]]

The [[area scale factor]] from the largest square to the middle one is $\frac{8}{18} = \frac{4}{9}$, so the [[length scale factor]] is $\frac{2}{3}$. This means that the point $E$ in the above diagram is two-thirds of the way along its side, and so by symmetry are $B$ and $D$. Then adding in the points at one-third along each side and joining as in the diagram divides the larger equilateral triangle into nine smaller equilateral triangles. The central equilateral triangle has area equivalent to three of these, so has area one third of the larger equilateral triangle.

The area of an [[equilateral triangle]] is $\frac{\sqrt{3}}{4}$ times the square of its side length, so the area of the larger equilateral triangle is $\frac{\sqrt{3}}{4} \times 18 = \frac{9\sqrt{3}}{2}$ and therefore the area of the inner equilateral triangle is $\frac{3\sqrt{3}}{2}$.



\end{document}