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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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## The fourth circle

Three circles and a line segment connecting three intersections. Prove that the three centres and the common intersection are concyclic.

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## Bicyclic transport

EHF is tangent to the purple circle at E, and EN is tangent to the red circle at N. A and C are the circle centers. A blue rectangle has three corners HEC and the point A on one side. What is the ratio of areas, green square to orange quadrilateral?

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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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## Rolling along

Tangents to the blue circle (ABC) intersect at a point K, and the line AK intersects (ABC) at H. D is the midpoint of BC. Show that the green circle (DHB) is tangent to the line AB.

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## Ten to nine

Two triangles and one incircle with its centre and tangency points. Prove that the triangles are similar.

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## Cathedral on the hills

A cathedral is erected on two hills, the side circular arcs whose centers are the hill ends, and so that the right (and left) side arcs are orthogonal. Show that the tip of the spire is directly above where the hills meet.

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## Tangent latitudes

Tangent lines QC and EC meet at C. A point D on QC has DC=1 and QD=2. The line ED intersects the circle at G, and the line HGI is parallel to QDC. What is HG/GI?

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## Square mouse

A semicircle and a square with extended side and diagonal. Prove that the red line segment is tangent to the semicircle.