### 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?

### Spirostars

A spiropath is a sequence of connected line segments end to end taking different directions. The same spiropath is iterated. When does it cycle and when does it go on indefinitely?

### Baby Circle

A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

# Orthogonal Circle

##### Age 16 to 18 Challenge Level:

 It is known that given any three non intersecting circles in the plane there is another circle or straight line that cuts the three given circles at right angles. (The circle or straight line is said to be orthogonal to the 3 original circles.) Given three circles with centres $(0, 0)$, $(3, 0)$ and $(9, 2)$ and radii $5$, $4$ and $6$ respectively find the centre and radius of the circle that cuts the three given circles at right angles. Draw the circles to check that the circle you have found appears to be orthogonal to the others.

What happens in the case of three circles with centres at $(0, 0)$, $(3, 3)$ and $(8, 8)$ and radii $1$, $2$ and $3$ respectively?

Given three circles, how can you tell without calculating which of the two cases applies, an orthogonal circle or an orthogonal straight line?