Four equilateral triangles. Two centres are shown. Prove that the green quadrilateral is a parallelogram.
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Triangles galore
Category: Advanced
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Folding magic
One corner of a rectangular piece of paper is folded as shown. Prove that the red line segments are concurrent.
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Lining up half way
Show that the three red midpoints are collinear.
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Baby steps
The edges of a cyclic quadrilateral extend to two intersections. O is the circle centre. The two diagonals intersect inside the circle. What is the angle α?
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The crosshairs
A regular pentagon and two semicircles. Prove that they are orthogonal, i.e. the tangents in the intersection point are orthogonal.
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The laser game
In an equilateral triangular mirror room a laser is shot from the top vertex down to the base. The light ray bounces three times to return to the same base point. What is the angle α?
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The basketball court
An orange circle is squeezed inside a rectangle. Show that no matter how the blue point moves on the circle, the derived points Q and P will satisfy |QL| = |KP|.
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The vanilla ice cream
A regular pentagon with two extended sides and a right angle. What is blue : green?
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Falling pillar II
Three congruent rectangles on a line. Prove that the four red points are cyclic.
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The paper airplane
Express the area of the orange triangle in terms of the areas of the other colours.