This problem also appears on Underground Mathematics, where you can find more materials to support classroom use.
Why do this problem?
This is a quick, simple problem on gradients with a neat result. It will help to reinforce ideas about lines and coordinate geometry and factorising expressions.
This problem is ideally used as a lesson starter. It might be useful when revisiting ideas about coordinate geometry in a more advanced context. The second part might challenge some students and could be left as an optional extra for those who find the first part straightforward.
In the problem, the result is described as 'beautiful'. Do students see it as such? Can they understand why a mathematician might see it as beautiful?
Can you find the gradients of the segments?
Can you find a similar result if the parabola were replaced by the cubic equation $y=x^3$.
How far can you repeat the analysis if two lines joining two pairs of points on the parabola were perpendicular?
Just do the first part of the problem.