A triangle with its incircle and three cevians. The tangency points and incentre are shown. Find the relation between the angles α, β and γ.
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Wheel of fortune II
Tag: circle
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In times out
What is the area of the green rectangle, which has sides tangent to the incircle and circumcircle of a triangle with side lengths a, b, c.? (Write the answer as a multiple of a quotient of two elementary symmetric functions in a, b, c.)
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The Bedouin tent
A semicircle and a triangle of which one side is tangent to the semicircle. What is the angle α?
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Shots in the dome
A beam from the top of one tower cuts through the dome, reflects off the ground, and hits the top of the other tower. Show that the line connecting the orange dots, the line connecting the blue dots, and the ground line are concurrent.
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Eclipsing earth
Two circles with their centres. The quadrilateral ABCD is cyclic and has two sides tangent to the small circle. Prove that BD is also tangent.
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Wedging the circle
Four similar triangles share one circle as incircle or excircle. Show that orange dotted lines must be concurrent.
A red equilateral triangle and its circumcircle. Prove that the blue triangle is also equilateral.
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The ceiling lamp
A regular octagon with two diagonals and a square. Prove that the four red points are concyclic.
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Interlocked triangles
Do the blue, green, and orange triangles have the same shape?
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Ball of yarn
A cyclic hexagon has concurrent cross diagonals. What is the ratio of the product of the orange sides compared to the product of the purple sides. (ace:bdf).