A regular pentagon and a right triangle. What’s the angle α?
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It’s all right II
Category: Advanced
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The three-body problem
Three coloured unit disks are placed so that they have a common (3-way) intersection, but none of the disks covers the intersection of the other two disks. What is the minimum radius of the disk that would cover all intersections at once, in all cases?
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The right position
A triangle, its incircle and a right triangle. The circle centre is shown in blue. Prove that the tangency points and right triangle vertex are collinear.
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Circle dancing
Start with a blue triangle, and form the green triangle whose vertices bisect each circular arc connecting blue vertices. Similarly, make the red triangle from the green, and the orange triangle from the red. Prove the triangle becomes equilateral in the limit.
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The prisoners dilemma
Two prisoners are chained to the walls of a triangular cell. The base is covered with a mirror and a perpendicular wall is separating them. They can each see the opposite corner through a hole. Proof they can see each other through another hole at the base of the perpendicular.
Note that this problem is known as the Blanchet Theorem.
From the purple triangle ABC and the center of its inscribed circle at D, form the three colored circles (BDC), (CDA), and (ADB), with centers F, G, H, respectively, thus forming the yellow triangle FGH. Prove that there is one circle that circumscribes both triangles.
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The maternal bond
Three tangent semicircles and a circle. One tangency point is shown. Prove that red : green = yellow : blue.
A triangle and two semicircles. What’s the angle α?
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Along similar lines
The blue triangle is similar to the green one and has the same orientation. The midpoints between corresponding vertices are connected to form a red triangle. Prove that it is also similar.
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The hawfinch II
A circle with two tangents. What’s red : blue in terms of α?