Assuming the two circles have centers on the x-axis, one C(x₁, r₁) is centered at (x₁, 0) and has the radius r₁, the other - C(x₂, r₂) - has the center at (x₂, 0) and radius r₂. The diagram shows the case of x₂ > x₁ and r₂ > r₁.

Quite clearly the sum of distances of the center O of the inscribed circle to the centers of the two given circles is r₁ + r₂, implying that the locus of the centers is indeed an ellipse with foci at the centers of the given circles and the major axis r₁ + r₂.

The minor axis can be found by considering the extreme case of the inscribed circle with the center at the top point of the ellipse. This point along with the centers of C(x₁, r₁) and C(x₂, r₂) form an isosceles triangle with sides (r₁ + r₂) / 2 and base (x₂ - x₁), making the altitude equal to

The minor axis of the ellipse is 2h.