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Intermediate

Ball of yarn

A cyclic hexagon has concurrent cross diagonals. What is the ratio of the product of the orange sides compared to the product of the purple sides. (ace:bdf).

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Advanced

Seeing double II

A right triangle is divided in two triangles by an altitude. The three incircles are shown with three tangency points. Prove that the two red line segments are congruent.

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Intermediate

The bike chain II

Two circles and four coloured common tangents. Prove they are congruent.

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Intermediate

The mid-circle

A blue right triangle has an orange-dotted circumcircle and a green incircle of radius 7. What is the radius of the purple circle, which is internally tangent to the orange circle and to the two sides of the right angle?

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Advanced

The open roof

A regular hexagon with two line segments at right angles. What is the minimum of yellow : red?

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Intermediate

The star shaft

A square with an equilateral triangle. M is the triangle side midpoint. What is yellow : red?

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Intermediate

Needles and yarn

The purple yarn is stretched between the coloured needles approximately as shown, and the yarn is repositioned to make the shortest path of that type. What is the length ratio purple/green?

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Intermediate

The brick oven

Five congruent rectangles. What’s their proportion width : height (aspect ratio)?

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Advanced

The mis-stacked trays

A rectangle is turned 90°, bisecting a bigger rectangle of the same shape. When squares are removed from the two side pieces, the remaining rectangles have the same shape as well. What is the common ratio of the rectangle side lengths?

Categories
Intermediate

Scratching the surface II

Two equilateral triangles, the smaller one being of variable size. What is the minimum value of orange : blue?