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

Given lines Lj , j = 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 Lj . In the case when the lines form the boundary of a four sided convex polygon R, let M1 and M2 be the midpoints of the diagonals of R. Let L be the line thru M1 and M2, let Z be the open line segment connecting M1 and M2, let Y be the closed line segment connecting M1 and M2, and let X be the open line segment which is the part of L lying inside R. It is well known that if an ellipse E is inscribed in R, then the center of E must lie on Z(see [1] and [2]). We prove(Theorem 11) that every point of Z is the center of some ellipse inscribed in R, which implies that the locus of centers of ellipses inscribed in R is precisely equal to Z. In addition, we prove(Theorem 11) that there is a hyperbola tangent to each of the Lj and with center (h, k) ∈ R if and only if (h, k) ∈ X − Y . More generally, any ellipse tangent to the Lj(and not just inscribed ones) must have its center on L. Introduction Given finitely lines Lj , j = 1, 2, 3, ..., N 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 Lj . Our main results concern four given lines (see §3), though we need results for two or three given tangents (see §1, §2, and §4). Most of the theorems extend easily to hyperbolas as well. It is useful to make the following definition. Definition. Given a finite set of distinct lines L1, L2, ..., LN in the plane, and an angle α, −2 < α < π 2 , suppose that there is an ellipse with center (h, k) and rotation angle α which is tangent to each of the Lj . Then we say that (h, k) is α admissible . If α = 0 we just call (h, k) admissible. We allow the angle of rotation to vary, and then look at the union, over α, of the α admissible centers, Sα. In the case when the lines form the boundary of a four sided convex polygon R, let M1 and M2 be the midpoints of the diagonals of R. Let L be the line thru M1 and M2, let Z be the open line segment connecting M1 and M2, let X be the open line segment which is the part of L lying inside R, and let Y equal the closed line segment connecting M1 and M2. It is well known that if an ellipse E is inscribed