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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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## 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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## The lorgnette

An ellipse, two circles centred in its focal points, a chord and a tangent. What is the angle α?

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## Pushing the boundary

Two equilateral triangles are placed side by side. What’s the proportion green : red?

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## The gyroscope

Two ellipses go through each others focal points. Prove that the focal points are the vertices of a parallelogram.

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## The Pappus eggs

Three semicircles and a circle with their centres. Two orange ellipses with focal points in the semicircle centres (they share the middle one). Prove that their upper intersection point coincides with the circle centre.

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