A triangle with two line segments. What’s the angle α?
Category: Advanced
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Angle chasing galore
Two intersecting circles with a triangle connecting them. Show that the three red points are collinear.
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Pink parallels
A pink parallelogram with two extended sides. What’s α : β?
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Ellipsoidal laser
The inside surface of an ellipse is a perfect mirror. There is a pin-hole at an end of the diameter. Show that a light ray emitted from either focus will exit the enclosure via the pin-hole, perhaps after bouncing through the foci several times.
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Roll over Beethoven
Two regular hexagons share a vertex. If the red hexagon has area 4, what is the area of the blue one?
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The hexagonal dart board
25 darts land inside a regular-hexagonal dart board with edge length the square root of 3. Show that at least five of the darts land inside the same unit circle.
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Wrapping presents
Six squares are wrapped together with red ribbons. Do they cross at a single point?
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The cheese dome
A semicircle and two tangent circles (tangency point shown). What is the angle α?
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The Cheshire Cat grin
What is the total area of the green teeth relative to the large circle area? (Assume the inner circle radius is 80% of the outer. The 32 lines forming the tooth sides are concurrent, and spaced at equal angles apart.)
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The high-water mark
Two equilateral triangles on a blue line segment. Prove that the red line segment is parallel to it.