A circle inscribed in a triangle inscribed in a semicircle. The line segment through two tangency points meets a radius at the semicircle circumference. What is the angle α?
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Blazing hot
Category: Advanced
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Camera obscura
A quarter circular room is covered with mirrors. A ray of light is emitted from one corner and reflected somewhere on the opposite wall, then on the arc, before reaching the adjacent wall at a point a distance a from the light source and b from the circle centre. What is the maximum of a/b?
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The freight elevator
Four points A, B, C and D form a quadrilateral. Prove that it is a rectangle if and only if for an arbitrary point P we have PA2+PC2=PB2+PD2.
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The Eightfold Path
A semicircle, a square and an isosceles triangle. What’s the angle α?
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Fun with flags
A triangle with two cevians. What is d/c-b/a?
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The Olympic bowl
A parabola is tangent to the equal sides of an isosceles triangle in its equal vertices. Another tangent is drawn as shown. Prove that the red line segments are congruent.
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As above, so below
A semicircle, a square, a triangle and two circles. A centre and a tangency point are highlighted. Prove that the circles are congruent.
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Mind the gap
A regular hexagon with an equilateral triangle sharing a vertex and having a vertex somewhere along the opposite side. What is the maximum length of line segment g?
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The crowded house
Three circles are inscribed in a rectangle. What is the area X in terms of A and B?
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The letter box
A rectangle with a diameter and two inscribed squares. What is its area in terms of the square areas A and B?