Explaining, convincing and proving

Can you work out the side length of a square that just touches the hypotenuse of a right angled triangle?

Three points A, B and C lie in this order on a line, and P is any point in the plane. Use the Cosine Rule to prove the following statement.

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

Can you create a Latin Square from multiples of a six digit number?

A 2-digit number is squared. When this 2-digit number is reversed and squared, the difference between the squares is also a square. What is the 2-digit number?

Which has the greatest area, a circle or a square, inscribed in an isosceles right angle triangle?

A napkin is folded so that a corner coincides with the midpoint of an opposite edge. Investigate the three triangles formed.

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

Find the missing angle between the two secants to the circle when the two angles at the centre subtended by the arcs created by the intersections of the secants and the circle are 50 and 120 degrees.

A hexagon, with sides alternately a and b units in length, is inscribed in a circle. How big is the radius of the circle?