Finding Ellipses and Hyperbolas Tangent to Two, Three, or Four Given Lines

Given lines , = 1, 2, 3, 4, in the plane, such that no three of the lines are parallel or are concurrent, we want to find the locus of centers of ellipses tangent to the . In the case when the lines form the boundary of a four sided convex polygon , let and be the midpoints of the diagonals of . Let be the line thru and , let be the open line segment connecting and , let be the closed line segment connecting and , and let be the open line segment which is the part of lying inside . It is well known that if an ellipse is in , then the center of must lie on (see [1] and [2]). We prove(Theorem 11) that every point of is the center of some ellipse inscribed in , which implies that the locus of centers of ellipses inscribed in is precisely equal to . In addition, we prove(Theorem 11) that there is a hyperbola tangent to each of the and with center () ∈ if and only if () ∈ − . More generally, any ellipse tangent to the (and not just inscribed ones) must have its center on