Two tangent semicircles inscribed in another semicircle. What is the total area?
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Cuckoo in the nest
Category: Advanced
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Five tangents
Five points ABCDE are equally spaced around the dotted circle. A smaller red circle is tangent to (ABCDE) at a point F, between B and C. Blue and green tangents are drawn from ABCDE to the red circle, and are coloured blue and green as shown. Show that the length sum of the blue tangents equals the length sum of the green tangents.
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Missing an angle
A triangle with three line segments to an internal point. What’s the angle α?
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Seven chords
A regular 7-gon inscribed in a circle (ABCDEFG), with some other point H on the circle, in the arc CD. Show that the length sum of the blue chords equals the length sum of the green chords.
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The mirror box
Light rays bounce off mirrored walls of a square before reaching one of the other corners. What are the slopes of the light rays leaving A that land in one of the corners B or D after exactly k=5 reflections?
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The gondola lift
A tangent and its normal in an arbitrary point on an ellipse with eccentricity e. The focal points are also shown. What is the minimum value of blue : red in terms of e?
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The grassy knoll
A triangle with three cevians that are concurrent in an arbitrary interior point. What is the maximum value of bdf/ace?
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A hyperbolic style
An arbitrary point B on the unit circle is reflected in the y-axis to give B’. D is the intersection of BC and AB’. Prove it lies on the red hyperbola.
Puzzle creator: Matthew Arcus.
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Shooting the moon
Show that the blue dots are concyclic and also the orange dots are concyclic.
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Drop shadow II
A right isosceles triangle contains another triangle. What is the angle α? Advanced if no trigonometry is used.