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# Regular Polygons

With equal angles and equal side lengths, what more could you want from a polygon?

# Regular Polygons - Decomposition into Triangles

Consider a rectangle with diagonals as shown in the above diagram. If the area of $$\triangle BCE$$ is $$32,$$ what is the area of $$\triangle ABE?$$

The above diagram is a regular pentagram. Let $$\lvert\triangle TRS\rvert$$ denote the area of $$\triangle TRS,$$ and $$\lvert\square PQRS\rvert$$ the area of $$\square PQRS.$$ If $$\lvert\triangle ABC\rvert=52,$$ what is the area of the shaded region $\lvert\square PQRS\rvert - \lvert\triangle TRS\rvert?$

In the above diagram, $$\square ABCD$$ is a square and $$\overline{DE}$$ is perpendicular to one of its diagonals $$\overline{AC}.$$ If the length of $$\overline{DE}$$ is $$21.$$ what is the length of $$\overline{AC}?$$

In the above diagram, figure $$ABCDE$$ is a regular pentagon centered at point $$F$$. Find the area of quadrilateral $$CDEF$$ when the area of triangle $$ABF$$ is $$56.$$

In $$\triangle ABC$$ above, $$L,M$$ and $$N$$ are the midpoints of $$\overline {AB}, \overline {BC}$$ and $$\overline {AC},$$ respectively. If the area of $$\triangle LMN$$ is $$54,$$ what is the area of $$\triangle ABC?$$

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