The vertices of a triangle lie on the hyperbola . Prove that its orthocentre also lies on the hyperbola.

**Solution**

We are working on the following figure

Let , and . Let us denote as its orthocentre. We have that:

Hence, the slope of the altitude is . Similarly,

Hence, the slope of the altitude is . Hence,

and

Solving this linear system we have

and finally . So, . This proves the claim.

This result is due to Poncelet and Brianchon.