Abstract: We will see a proof of Pythagoras theorem on the plane and another on the sphere. Consider the paths traced by the front and rear wheel of a bicycle that returns to its initial position. We will discuss an interesting fact about the signed area between the tracks. If time permits, we will discuss another result on the sphere relating angles and area of a triangle. (Message to the experts: the theme of the talk will be to introduce the audience to the geometry around the Gauss-Bonnet theorem).