### 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.

### Possible approach

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?

### Key question

Can you find the gradients of the segments?

### Possible extension

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?

### Possible support

Just do the first part of the problem.