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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Eclipsing earth
Category: Intermediate
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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.
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Interlocked triangles
Do the blue, green, and orange triangles have the same shape?
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Six four
A regular hexagon and an inscribed tilted rectangle. What is the angle α?
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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).
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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.
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The sun hat
Two triangles share a circumcircle and vertex, with one edge of the orange triangle containing the feet of two of the altitudes of the blue triangle. Show that the orange triangle is isosceles.
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Uneasy the head
The green zigzag crown segments would extend through either B or C. Show that the arcs along the top are equally spaced.
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Giant’s shoulder II
Three equilateral triangles share three vertices. What is blue : red?