Can you make sense of these three proofs of Pythagoras' Theorem?

What is the same and what is different about these circle questions? What connections can you make?

Can you sort these triangles into three different families and explain how you did it?

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?

Can you make sense of the three methods to work out the area of the kite in the square?

Make your own pinhole camera for safe observation of the sun, and find out how it works.

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

Anamorphic art is used to create intriguing illusions - can you work out how it is done?

The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

The diagonals of a square meet at O. The bisector of angle OAB meets BO and BC at N and P respectively. The length of NO is 24. How long is PC?

Try out this geometry problem involving trigonometry and number theory