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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Pop you bubble
Tag: triangle
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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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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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Interlocked triangles
Do the blue, green, and orange triangles have the same shape?
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Seeing double II
A right triangle is divided in two triangles by an altitude. The three incircles are shown with three tangency points. Prove that the two red line segments are congruent.
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Sunrise over green mountain
An acute triangle mountain ABC has altitudes BE and CF. The dotted tangent lines to the sunny circumcircle (AEF) at E and F intersect at a point M. Show that M is on the mountain’s base BC.
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The silly walk
A circle and several triangles. Prove that the green triangle is isosceles.