Final answer:
If the sum of the tangents of the base angles of a triangle on a given base is constant, then the locus of the vertex is an ellipse.
Step-by-step explanation:
An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci is a constant. This property is the defining characteristic of an ellipse. Therefore, if a triangle is described on a given base such that the sum of the tangents of the base angles is constant, then the locus of the vertex of the triangle is an ellipse.