Sine, cosine, tangent

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

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

Draw three equal line segments in a unit circle to divide the circle into four parts of equal area.

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.

There are many different methods to solve this geometrical problem - how many can you find?

A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

Can you deduce the familiar properties of the sine and cosine functions starting from these three different mathematical representations?

How would you design the tiering of seats in a stadium so that all spectators have a good view?

Three squares are drawn on the sides of a triangle ABC. Their areas are respectively 18 000, 20 000 and 26 000 square centimetres. If the outer vertices of the squares are joined, three more triangular areas are enclosed. What is the area of this convex hexagon?

Prove Pythagoras' Theorem for right-angled spherical triangles.