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Given 3 points A, B, C, define G so CG=CB and angle GCB is 120 degrees. Define M as the midpoint of side CH of the parallelogram CGAH. The green equilateral has side CM and the purple equilateral has side MB. Is AJK also equilateral?

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## Hidden parallelogram

Blue circle (ACDGB) with CEG || AB. Pink circle (EDG) intersects AB at I. Extend DI to intersect blue circle at J, then JC intersects AB at M. Show that IM = CE.

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## The square sail

A quadrilateral and a parallelogram share two sides, and a vertex from each determine the orange square. The lengths BC, EF, AD are 4,5,7 respectively, where E and F are midpoints of AB and DC. What is the orange area?

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## Euclid’s wing

Several line segments, some of which are parallel. What is green : orange?

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## Dancing squares

Four squares and one centre. Prove that the red quadrilateral is a parallelogram.

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## Triangles galore

Four equilateral triangles. Two centres are shown. Prove that the green quadrilateral is a parallelogram.

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## The lighthouse

A parallelogram and four equilateral triangles. What is yellow : green?

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## Pink parallels

A pink parallelogram with two extended sides. What’s α : β?

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## Three’s a crowd

Four equilateral triangles. Prove that the red quadrilateral is a parallelogram.

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## The gyroscope

Two ellipses go through each others focal points. Prove that the focal points are the vertices of a parallelogram.