# Equal Areas in Equilateral Triangle

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A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet purports to illustrate one of the many properties of an equilateral triangle.

Let M be a point inside an equilateral triangle ABC, with pedal points A', B', and C', as shown. The lines joining M to the six points A, B, C, A', B', C' split the triangle into six smaller ones. The applet suggests that

(1)

Area(AMB') + Area(BMC') + Area(CMA') = Area(A'MB) + Area(B'MC) + Area(C'MA).

(I am grateful to Manuel Silva from Portugal for bringing this problem to my attention.)

As in a sister problem, Draw A_{b}A_{c}||BC, B_{a}B_{c}||AC, and C_{a}C_{b}||AB. The three lines split ΔABC into three parallelograms and three smaller equilateral triangles. The three parallelograms are divided by the diagonals joining M to the vertices A, B, C into equal parts, whereas the three equilateral triangles are divided into equal parts by the lines joining M to the pedal points A', B', C'. So, for a proof, one only needs to rearrange the areas in (1).

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