ABC is an acute-angled scalene triangle

In this case we have the hyperbola for the first time. The question to be asked is can we predict through wich of the vertices the hyperbola will enter and leave and at which vertex will it simply touch?

Claim: The hyperbola passes through the vertices at which the triangle has the largest and smallest angles while touching at the vertex at which the middle sized angle occurs.

To test the claim and watch a GSP sketch draw the locus click here.