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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?

### Flower

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 ${\angle APB}$ ?