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Flexi Quads

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Lunar Angles

What is the sum of the angles of a triangle whose sides are circular arcs on a flat surface? What if the triangle is on the surface of a sphere?


Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.

Square World

Age 16 to 18 Challenge Level:

$P$ is a point inside a square $ABCD$
such that $PA= 1$, $PB = 2$ and $PC = 3$.

How big is ${\angle APB} $ ?