### Pericut

Two semicircle sit on the diameter of a semicircle centre O of twice their radius. Lines through O divide the perimeter into two parts. What can you say about the lengths of these two parts?

### 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.

# Baby Circle

##### Stage: 5 Challenge Level:

Rosalind of Madras College sent in this solution, well done Rosalind.

All three circles touch each other and have a common tangent $DFE$. The radius of the baby circle is $r$ and the radii of the other circles are $BE = 1$ unit and $AD = 2$ units. So $AC = 1$ unit, $AB = 3$ units, $AH = 2 - r$, $AR = 2 +r$, $BR = 1 + r$ and $BG = 1 - r$. Using Pythagoras Theorem: \eqalign{ \; RG &=& \sqrt{(1+r)^2 - (1-r)^2} = \sqrt{4r} = 2\sqrt{r} \\ \; HR &=& \sqrt{(2+r)^2 - (2-r)^2} = \sqrt{8r} = 2\sqrt{2r} \\ \; CB &=& 2\sqrt{2} = DE \\ RG + HR = CB &\Rightarrow& 2\sqrt{r} + 2\sqrt{2r} = 2\sqrt{2}.} This gives $\sqrt{r}(1+\sqrt{2}) = \sqrt{2}$ and hence by squaring \eqalign{ r &=& \frac{2}{3+2\sqrt{2}} \\ \; &=& \frac{2(3 - 2\sqrt{2})}{(3+2\sqrt{2})(3-2\sqrt{2})} \\ \; &=& \frac{6 - 4\sqrt{2}}{9 - 4\times2} \\ \; &=& 6 - 4\sqrt{2}} So the radius of the baby circle is $6 - 4\sqrt{2}$.