The vertices of a triangle lie on the hyperbola . Prove that its orthocentre also lies on the hyperbola.
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,
Solving this linear system we have
and finally . So, . This proves the claim.