prove that the circle drawn on any focal distance as diameter touches the auxiliary circle in an ellipse

Prove that the circle described on the focal chord as a diameter touches the directrix

Circle described on focal length as diameter always touches the auxilliary circle.

What is Auxiliary circle?

Circle described on the focal chord as diameter touches the directrix.

The tangent at a point P on an ellipse intersects the major axis at T, and N is the foot of the perpendicular from P to the same axis.Show that the circle drawn on NT as diameter intersects the auxiliary circle orthogonally.

Circle described on the focal chord as diameter touches the tangent at the vertex

Prove that the circle drawn with any side of a rhombus as a diameter,posses through the point of intersection of its diagonals.

Prove that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.

Prove that the circle drawn on any one of the equal sides of an isosceles triangle as diameter bisects the base.