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.

Shape and Territory

If for any triangle ABC tan(A - B) + tan(B - C) + tan(C - A) = 0 what can you say about the triangle?

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

Three by One

Stage: 5 Challenge Level:

Here is a hint to help on you on the way :

Let $\alpha + \beta = \gamma$ and $\tan(\alpha + \beta) = \frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}$

where $\alpha = \tan^{-1}{1\over3}$, $\beta = \tan^{-1}{1\over2}$, $\gamma = \tan^{-1}1$.