A circle and several triangles. Prove that the green triangle is isosceles.
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The silly walk
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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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Video: Peeping out
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Stay centred
Two squares share a vertex. Prove that the red point is the circumcentre of the red triangle.
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Video: The bike chain II
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Peeping out II
A unit square and a rectangle touching three square sides and passing through the midpoint of the upper side. What is the minimal blue area?
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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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Video: Uneven brickwork
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Giant’s shoulder II
Three equilateral triangles share three vertices. What is blue : red?
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Falling in
Start with an acute triangle and form a new triangle from the points of tangency of its inscribed circle. Continue this process to make make the triangle with blue vertices. What is the maximum possible angle at a blue vertex?