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## Middle journey

The midpoint (center) of the incircle is I. R is the midpoint of the red segment and G is the midpoint of the green segment. Show the RIG makes a line.

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## Tip of tipi

A regular hexagon is divided in four triangles and one quadrilateral. Express the hexagon area in terms of areas A, B and C.

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## Pentagon pleasure II

A regular pentagon and its circumcircle. What is blue : orange?

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## The fountain plaza

The incircle of ABC touches sides at I and J. D, E, F are the bases of altitudes from C, B, and A. The incenters of BDF of CEF are N and M, respectively. Show that JIMN is a parallelogram and that IM is perpendicular to BC.

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## Man in the middle

Three equilateral triangles share a vertex. What is the area of the blue one in terms of green and red?

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## Shoulder to shoulder

Two equilateral triangles sharing a vertex and several line segments. What’s the angle α?

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## Locus pocus II

What is the locus of points P for which perpendicular lengths (red=l, green=n, purple=m) to the sides of an isosceles triangle satisfy the condition red*green = purple2?

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## The center falls apart

In a right triangle ACB show that the incircle touches sides AC and CB in points that are vertically lined up with the centers of the incircles for ADC and BDC. Furthermore, show that the red and green distances are equal.

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## Conway circular area

A triangle with sides a, b, c, when extended to whiskers of opposite side length, forms the “Conway Circle”. What is the area of the circle in terms of the expressions a+b+c, ab+bc+ca, abc of the side lengths a, b, c.

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## Family constellation

Two equilateral triangles on a line. Prove that the blue triangle is equilateral as well.