### 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?

### Kissing

Two perpendicular lines are tangential to two identical circles that touch. What is the largest circle that can be placed in between the two lines and the two circles and how would you construct it?

### Logosquares

Ten squares form regular rings either with adjacent or opposite vertices touching. Calculate the inner and outer radii of the rings that surround the squares.

# Orthogonal Circle

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

Call the centre of the orthogonal circle $(h, k)$ and the radius $r$ (equal to $CA$ in triangle $ABC$). Use Pythagoras' Theorem for triangle $ABC$ to write down an equation. In a similar way find equations using the other two circles.