### 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

### Strange Rectangle 2

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

# Shape and Territory

##### Stage: 5 Challenge Level:

Sue Liu, S5, Madras College sent in a good solution which shows that if $A, B$ and $C$ are angles in a triangle and $$\tan (A - B) + \tan (B - C) + \tan (C - A) = 0$$ then the triangle is isosceles. Can you prove a stronger result? We start with the expression $$\tan (A - B) + \tan (B - C) + \tan (C - A) = 0.$$ Write $X = A - C$ and $Y = B - C$, then the given expression becomes $$\tan (X - Y) + \tan Y + \tan -X = 0.$$ This gives $$\tan (X - Y) = \tan X - \tan Y$$ and we know the identity $$\tan (X - Y) = {{\tan X - \tan Y}\over {1 - \tan X \tan Y}}.$$ Hence either $$\tan X = \tan Y \quad (1)$$ or $$\tan X \tan Y = 0 \quad (2)$$ In case (1) we show that the angles $X$ and $Y$ are equal. $$|X - Y| = |A - B| < A + B < 180 ^\circ$$ and the tan function is periodic with period 180 degrees so $X = Y.$ This gives $A - C = B - C$ hence $A = B$, so the triangle is isosceles. In case (2), either $\tan X = 0$ or $\tan Y = 0$, hence $A = C$ or $B = C$ and in all the cases the triangle is isosceles.