### 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.

### So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

### Why Stop at Three by One

Beautiful mathematics. Two 18 year old students gave eight different proofs of one result then generalised it from the 3 by 1 case to the n by 1 case and proved the general result.

# Shape and Territory

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

If for a triangle $ABC$

$\tan(A - B) + \tan(B - C) + \tan(C - A) = 0$

what can you say about the triangle?