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The sliding square

A square of variable size inscribed in a semicircle of radius r. What is the maximum distance d of the lowest square vertex to the nearest semicircle corner?

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The turning point

Two isosceles triangles share a vertex. What is the maximum of their area proportion red/green?

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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 Queen’s chamber

A right triangle with three squares forming a smaller inner triangle. What is the maximal blue fraction?

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The ocean view

Two squares of variable size share a vertex. What is the maximal blue fraction?

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The Bermuda Triangle

A triangle with an inscribed rectangle. What is the maximum red fraction?

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The rolling stones

If you roll a circle down a parabola, small circles can roll all the way down and up, whereas large circles get stuck between the sides at some point. What is the transition radius? In other words, what’s the maximum circle that keeps on rolling?

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The PowerPoint

A point lies 8 from the centre of a circle of radius 6. The angle α is variable. What’s the maximum triangle area?

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Don Quixote and the windmill

Don Quixote is approaching a windmill. The sails of length √2 are attached to a point that is 2 above eye level. What is the maximum vertical angle that Don Quixote perceives as he moves forward?

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Intermediate

Beware of the dots

Two congruent circles and a common tangent. A triangle connecting tangency points and the upper intersection point. What angle α maximises its area?