An obtuse triangle, of which α is the obtuse angle, with its incircle, two cevians and two tangents. Find a relation between α and β.
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In and out
Tag: circle
Two quarter-circles fit inside a square, and five circles fit inside the overlapped region, centred. What is the ratio of the (straight line) segment lengths red to blue? (The segments connect to points of tangency.)
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Pop you bubble
A square with a semicircle and a circle of equal radius. Their tangency point is shown. Prove that the red triangle is equilateral.
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Wheel of fortune II
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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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.