Implications of Marden s Theorem for Inscribed Ellipses

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1 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 1 Implications of Marden s Theorem for Inscribed Ellipses Mahesh Agarwal, John Clifford and Michael Lachance Abstract. We give a constructive proof for the existence of inscribed family of ellipses in convex n-gons for 3 n 5 using Marden s theorem. In the case of a pentagon, we also show how Marden s theorem can be made to exhibit simultaneously two ellipses, one inscribed in the pentagon and the other inscribed in its diagonal pentagon. The two ellipses are as intrinsically linked as are the pentagon and its diagonal pentagon. Our method uses the theory of dual curves. 1. INTRODUCTION The goal of this paper is to show how Marden s Theorem (Theorem 4.2) in [8] lends itself to investigating families of inscribed ellipses in convex n-gons. Marden s Theorem, a refinement of the Gauss-Lucas Theorem [8], states that the non-trivial critical points of the product of complex power functions p(z) n k1 (z z k) m k for m k R, m k 0 are the foci of a curve of class n 1 that touches the side z j z k of the n-gon formed by z 1, z 2,, z n, in the ratio m j : m k. Kalman [6] and Minda and Phelps [9] rejuvenated interest in Marden s Theorem in their Monthly article An Elementary Proof of Marden s Theorem and Triangles, ellipses, and cubic polynomials respectively. Some of theses articles are [2, 3, 5, 4]. The first proofs of Marden s Theorem date back to Siebeck [10] in 1864, Bôcher [1] in 1892, and Linfield [7] in 1920, to name a few. We are following Kalman s lead in naming the theorem Marden s Theorem. More on the history of Marden s Theorem can be found in Marden s book [8]. In [6] Kalman provides an elegant proof of a particular case of Marden s Theorem: given a triangle with vertices z 1, z 2, z 3, there is a unique inscribed ellipse that is tangential to all three sides and the point of tangency divides each side in the ratio 1 : 1. He does this by considering a polynomial p(z) (z z 1 )(z z 2 )(z z 3 ) and showing that the critical points of this cubic are the foci of the ellipse that is tangential to the three sides at their respective midpoints. In [7] Linfield proves that the nontrivial critical points of p(z) (z z 1 ) m 1 (z z 2 ) m 2 (z z 3 ) m 3 are the foci of the ellipse inscribed in the triangle formed by the zeros of p(z) and tangent to the sides of the triangle in the ratio m j : m k. An immediate consequence of his work is the existence of a two parameter family of ellipses that are tangential to the sides of the triangle formed by the vertices z 1, z 2, z 3. In this paper, we completely characterize ellipses that can inscribed in strictly convex n-gons. The main theorem of the paper is the following: Theorem 1 (Main Theorem). Concerning inscribed ellipses in convex nondegenerate n-gons: In triangles there exists a unique two-parameter family of inscribed ellipses; In quadrilaterals there exists a unique one-parameter family of inscribed ellipses; In pentagons there exists precisely one inscribed ellipse; For n 6, there exist n-gons for which there are no inscribed ellipses; whenever there is an inscribed ellipse, it is unique. January 2014] MARDEN S THEOREM FOR INSCRIBED ELLIPSES 1

2 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 2 We recall the precise statement of Marden s theorem. Theorem 2 (Marden s Theorem). (Theorem 4.2 in [8]) The zeros of the function F (z) n k1 m k z z k, m k R, m k 0, are the foci of the curve of class n 1 that touches each line segment z j z k in a point dividing the line segment in the ratio m j : m k. Remark. The function F (z) is the logarithmic derivative of p(z) n k1 (z z k ) m k, that is, F (z) d dz log p(z) p (z) p(z). Its zeros describe the nontrivial critical points of the polynomial p(z) if m k s are positive integers. z 4 Figure 1. Illustration of Marden s theorem for n 4 case Example. Figure 1 is an illustration of Marden s theorem for the polynomial p(z) (z 2 + 1)(z + 2)(z 1). Definition. The number of tangent lines to a curve through a generic point is invariant and is called the class of the curve. Remark. Plücker defined the foci of a curve to be the real intersections of the tangent lines to it drawn through the homogenous points at infinity [1 : ±i : 0] (see section 2). In the case of degree two curves, this definition reduces to the traditional one for conics. Since our work concerns ellipses, the more general notion of foci is not necessary for our exposition. Remark. The existence of the elliptical families in the Main Theorem follows from: 2 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121

3 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 3 r r 1 r 1 r s 1 s Figure 2. Implications of Brianchon s Theorem for inscribed ellipses Theorem 3 (Brianchon s Theorem). If a hexagon circumscribes an ellipse, then its three diagonals meet in a point; conversely, if the three diagonals of a hexagon meet in a point, then there exists an inscribed ellipse. To see that there exists a two-parameter family of ellipses inscribed in a triangle, fix 0 < r < 1 and 0 < s < 1 as in the triangle in Figure 2. The five solid dots and one open dot form vertices of a degenerate hexagon; these vertices satisfy the hypothesis of the converse of Brianchon s Theorem, guaranteeing the existence of an inscribed ellipse. For an ellipse to be tangent to the six sides of the hexagon, it must be tangent at the points designated by r, s, and the open dot. The fact that r and s are arbitrary explains the existence of a two-parameter family. Similar arguments can be made for the quadrilateral and pentagonal cases above. No inscribed ellipse exists for the hexagon in Figure 2. Remark. Brianchon s theorem establishes the existence of the families of ellipses, but does not offer a means for exhibiting the ellipses explicitly. This we shall do using Marden s theorem. 2. DUALITY AND DUAL CURVES In this section we fix notations and definitions that will be used for the rest of the paper. For more details we refer the reader to [11]. Let R be the field of real numbers and P 2 the real projective plane. To each point (x, y) in R 2 we associate a homogeneous point [x : y : 1] in P 2, and to each line ax + by + c 0 in R 2 we associate a homogeneous line ax + by + ch 0 in P 2. Just as λax + λby + λch 0 for λ 0 identifies the same line ax + by + ch 0, so it is understood that [λx : λy : λ] [x : y : 1] for λ 0. That is, it is the ratio between components that unambiguously distinguishes among homogeneous points. The homogeneous point [b : a : 0] can be seen to lie on the homogeneous line ax + by + ch 0, and is referred to as a point at infinity. All the points at infinity lie January 2014] MARDEN S THEOREM FOR INSCRIBED ELLIPSES 3

4 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 4 on the homogeneous line at infinity, h 0. To each homogeneous point Q and line L P 0 in the xyh-plane there exists a unique line L Q 0 and point P, respectively, in a dual αβγ-plane. Specifically, Q [x : y : h] L Q : Q (α, β, γ) 0 L P : P (x, y, h) 0 P [α : β : γ], where L Q : Q (α, β, γ) xα + yβ + hγ and similarly, L P : P (x, y, h) αx + βy + γh. Note that L Q is a linear homogeneous polynomial in the variables (α, β, γ), while L P is a linear homogeneous polynomial in the variables (x, y, h). The point Q and line L Q 0 are duals of one another; as are P and L P 0. Note that L P P, so that the dual of a homogeneous line can be seen to be its gradient. In general, to any homogenous curve ϕ ϕ(α, β, γ) P 2 of degree n 1, n 1, we can associate a homogenous dual curve ϕ ϕ(x, y, h) P 2, whose generic points are tangents to ϕ P 2. Symbolically, ϕ {[x : y : h] P (x, y, h) 0, ϕ(p ) 0}. So the duals of the points of ϕ 0 are tangent lines to ϕ 0; analogously, tangent lines to ϕ 0 are duals of the points of ϕ 0. To determine the points of ϕ 0 we require the set of tangent lines to ϕ 0. To this end the homogeneous tangent line to the curve ϕ 0 at any point P 0 on the curve can be written as ϕ(p 0 ) (P P 0 ) 0. According to Euler s homogeneous function theorem, ϕ(p ) P (n 1)ϕ(P ), simplifying the representation of the tangent line: ϕ(p 0 ) P 0. Thus if P 0 lies on ϕ 0, then Q 0 ϕ(p 0 ) lies on ϕ 0. Remark (Explicit formula for a dual of a conic). If ϕ is quadratic then (x, y, h) φ(α, β, γ) is a linear system. If α α(x, y, h), β β(x, y, h), and γ γ(x, y, h) is a solution to this system then the dual conic is given by ϕ(x, y, h) ϕ(α(x, y, h), β(x, y, h), γ(x, y, h)). The class and degree of a conic is always two. The class and degree is not generally equal for curves of degree greater than two. Example. Figure 3 is an illustration of a cubic polynomial ϕ(α, β, γ) α(α γ) 2 β 2 γ and its fourth degree dual ϕ(x, y, h) 4y 4 + x 2 y 2 18xy 2 h 27y 2 h 2 4x 3 h. The dual of a point P 0 lying on ϕ 0 is the line P 0 (x, y, h) 0 tangent to ϕ 0 at Q 0, and vice versa. Corollary 1 (Marden s Polygon Corollary). Let P denote a convex n-gon, with vertices labeled V k, k 1,..., n and let m k, k 1,..., n be positive real numbers. Then there exists a class n 1 curve, which is tangent to the n(n 1)/2 lines joining the vertices of P with the points of tangency given by: m j m k V k + V j. (1) m j + m k m j + m k 4 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121

5 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 5 4y 4 x 2 y 2 18xy 2 h27y 2 h 2 4x 3 h0 ΑΑΓ 2 Β 2 Γ0 Q 0 P 0 P 0 x,y,h0 Q 0 Α,Β,Γ0 Figure 3. Dual homogeneous curves Proof. Following Linfield [7], we associate to each vertex V k (x k, y k ) of the polygon a homogeneous point and homogeneous linear polynomial Q k [x k : y k : 1] L k Q k [α : β : γ] x k α + y k β + γ. The intersection point P jk of the lines L j 0 and L k 0 is either finite or a point at infinity: P jk { [y j y k : x k x j : x j y k x k y j ], L j 0, L k 0 not parallel; [y k : x k : 0], L j 0, L k 0 parallel. The dual of the point P jk is the line joining Q j to Q k. Let ϕ n m l L 1 L l 1 L l+1 L n (2) l1 The homogeneous polynomial ϕ is of degree n 1 and interpolates P jk since each summand in its definition contains as a factor either L j or L k. Let ϕ 0 denote the dual to the curve ϕ 0. The fact that ϕ is of degree n 1 implies that ϕ is of class n 1. By definition, ϕ 0 is tangent to the line joining Q j and Q k, since ϕ interpolates the dual of this line, P jk. To determine the point of tangency, Q jk, we compute the tangent line to ϕ 0 at P jk. Since each of the summands of ϕ contain at least one if not both of the factors L j and L k, it can be expressed as ϕ (m j L k + m k L j )A jk + L j L k B jk for appropriate A jk and B jk. Thus ϕ(p jk ) A jk (P jk )(m j Q k + m k Q j ). Normalizing the homogenous component of ϕ(p jk ) we see that the dual of the tangent line to ϕ 0 at P jk corresponds to Q jk : m j m k Q k + Q j, m j + m k m j + m k January 2014] MARDEN S THEOREM FOR INSCRIBED ELLIPSES 5

6 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 6 establishing (1). Definition. The function ϕ defined in (2) plays a critical role in what follows. Hereafter we shall refer to it as Marden s function. 3. PROOF OF MAIN THEOREM In the case of a triangle (n 3), Marden s function ϕ is a quadratic, and hence its dual ϕ must also be a quadratic. We exhibit a simple set of constraints on the exponents m k, that depend upon two parameters, that compel the de-homogenized quadratic to be an ellipse. In the case of a convex quadrilateral (n 4), Marden s function ϕ is a cubic. We produce a set of exponents, depending upon a single parameter, that enable ϕ to be factored into the product of a linear and quadratic polynomial. Its de-homogenized dual, ϕ, in this instance is a point and an inscribed ellipse. For a convex pentagon (n 5), ϕ is a quartic. There exists a unique set of constraints on the exponents that permit ϕ to be factored into the product of two quadratic polynomials. The de-homogenized duals of these factors will be shown to be inscribed in the pentagon and in the diagonal pentagon, respectively. In triangles there exists a unique two-parameter family of inscribed ellipses. Proof. Let T denote a nondegenerate triangle with homogeneous vertices Q k, k 1, 2, 3, labeled clockwise. Marden s function is a quadratic in term of α, β and γ. ϕ m 1 L 2 L 3 + m 2 L 1 L 3 + m 3 L 1 L 2 ΑΒΓ Q 1 r P 12 x,y,h 0 L 3 0 Q 12 1 r P 23 Φ 0 Q 2 Φ 0 P 13 L 1 0 Q 3 P 12 L 2 0 Q 12 Α,Β,Γ 0 Figure 4. Illustration of Marden s theorem for n 3 To establish the existence of the two-parameter family of ellipses inscribed in T, fix two parameters 0 < r, s < 1. Geometrically, these values designate two points Q 12 (1 r)q 1 + rq 2 and Q 23 (1 s)q 2 + sq 3 on adjacent sides of T (see Figure 1). Choose positive m 1, m 2 and m 3 so that m 1 m 2 r 1 r, and m 2 m 3 s 1 s. 6 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121

7 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 7 Then the dual curve ϕ 0 is also quadratic, a conic upon de-homogenization. Since the curve is tangent to the three segments, and all m i s are positive, it follows that ϕ 0 must be an ellipse. There are no other ellipses inscribed in the triangle other than those of the twoparameter family. If such an ellipse did exist, then it would be tangent to the sides Q 1 Q 2, Q 2 Q 3, and Q 1 Q 3 ; the points of contact on the first two sides define parameters r 0, s 0. The equation of this hypothesized ellipse satisfies five independent homogenous constraints in common with the de-homogenized dual ϕ for this choice of parameters r 0, s 0. Hence the ellipse is among the two-parameter family. In convex quadrilaterals there exists a one-parameter family of inscribed ellipses. Proof. Let Q denote a convex nondegenerate quadrilateral with homogeneous vertices Q k, k 0,..., 3, labeled clockwise. In this instance Marden s function ϕ m 0 L 1 L 2 L 3 + m 1 L 0 L 2 L 2 + m 2 L 0 L 1 L 3 + m 3 L 0 L 1 L 2 (3) is cubic. Without loss of generalization we assume that the intersection of the diagonals of Q lies at the origin and that the vertices Q 1 and Q 3 lie on the vertical axis. Thus there exist constants 0 < θ, φ < 1 for which (1 θ)q 1 + θq 3 [0 : 0 : 1] and (1 φ)q 2 + φq 0 [0 : 0 : 1]. (4) For a parameter 0 < r < 1. Choose positive real numbers m 0, m 1, m 2 and m 3 such that m 1 m 2 r 1 r, m 1 m 3 θ 1 θ, and m 2 m 0 φ 1 φ. (5) Conditions (4) imply (1 θ)l 1 + θl 3 γ (1 φ)l 2 + φl 0. So, upon writing (3) as ϕ ( m 2 L 0 + m 0 L 2 ) L1 L 3 + ( m 1 L 3 + m 3 L 1 ) L2 L 0, and taking into account the constraints (5), it can be verified that Marden s function can be factored into a linear term times a quadratic one: γ ( ) (1 r)θl1 L 3 + rφl 2 L 0. (6) (1 r)θφ The dual of the linear factor γ/((1 r)θφ) is the origin; the dual of the quadratic is a conic. Consequently, the dual of ϕ is a conic, tangent to the interior of the four segments defining Q, and hence must be an ellipse. As in the case of the triangle, any ellipse inscribed in the quadrilateral would necessarily share five independent conditions with ϕ for a particular choice of r. Hence this family is unique. Remark. The linear factor in the function ϕ in (6) corresponds to the line at infinity because we constrained the intersection of the diagonals of Q to lie at the origin. If this were not the case, then the linear factor of ϕ would correspond to the dual of that point of intersection. Remark. Since we can translate and rotate a quadrilateral Q so that it fulfills the conditions in the above proof, we can apply the rotation and translation to the computed two-parameter family in reverse to obtain the desired family inscribed in Q. January 2014] MARDEN S THEOREM FOR INSCRIBED ELLIPSES 7

8 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 8 In pentagons there exists a unique inscribed ellipse. Proof. Let P denote a convex nondegenerate pentagon. Extend two of the sides of P to form a convex quadrilateral Q from which the pentagon can be thought to have been cut (see Figure 5). Label the vertices of Q in a clockwise fashion Q k, for k 0, 1, 2, 3, beginning with the vertex not among those of P. Label the remaining vertices of P consecutively Q 4 and Q 5. Without loss of generalization we assume that, as in the n 4 case, the diagonals of Q intersect at the origin, and additionally that the vertices Q 1 and Q 3 lie on the vertical axis (see Figure 5). With this structure in place there exist constants 0 < θ, φ < 1 for which (1 θ)q 1 + θq 3 0 and (1 φ)q 0 + φq 2 0. Note that in Figure 5, we suppress the labeling of the lines that are the duals of the vertices of the pentagon and its diagonal pentagon to simplify the image. Note that the lines can be inferred from the labeled intersection points. For example, P 015 lies on the three lines L 0 0, L 1 0, and L 5 0. The coincidence of these three lines follows from the observation that the vertices Q 0, Q 1, and Q 5 are collinear. Q 1 Q 5 P C24 Φ DP 0 Q A Φ P P 0 23 P 034 P BC35 Q 0 Q B 0 Φ DP 0 Q 2 0:0:1 P 45 Q 4 Q C Φ P 0 P 12 P 015 P AB14 P A25 Q 3 Figure 5. Illustration of Marden s theorem for n 5 Marden s function in this case takes the form ϕ m 1 L 2 L 3 L 4 L 5 + m 2 L 1 L 3 L 4 L 5 + m 3 L 1 L 2 L 4 L 5 + m 4 L 1 L 2 L 3 L 5 + m 5 L 1 L 2 L 3 L 4. We shall show that ϕ can be expressed as the product of two quadratic polynomials for appropriate choices of the exponents m 1,..., m 5. The first factor of ϕ is modeled on the quadratic factor in (6), ϕ P (1 r)θl 1 L 3 + rφl 0 L 2. (7) Since each L k vanishes at any point with k among its subscript, ϕ P 0 can be seen to interpolate the four points P 12, P 23, P 034, and P 015 for any choice of the free parameter r. We now show r in (7) can be chosen so that ϕ P also interpolates a fifth point, P c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121

9 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 9 Once we have shown that ϕ P 0 interpolates the duals of the five sides of P, it follows immediately that ϕ P 0 must be tangent to the sides of P. Because Q 0 and Q 1 are multiples of Q 2 and Q 3, respectively, it follows that L 0 0 and L 2 0 are parallel, as are L 1 0 and L 3 0. Furthermore, since Q 1 and Q 3 are assumed to lie on the vertical axis, L 1 0 and L 3 0 must be parallel to the α-axis, and intersect at P 13 [1 : 0 : 0]. Because the dual image of a strictly convex polygon is also strictly convex, the point P 45 must lie between L 1 0 and L 3 0, and to one side of L 0 0 and L 2 0 (see Figure 5). Since each of the homogeneous linear polynomials L 0, L 1, L 2, and L 3 is positive at the origin [0 : 0 : 1], it follows that the product L 1 L 3 at P 45 is positive, and the product L 0 L 2 at P 45 is negative. Thus there exists a unique parameter value 0 < r 0 < 1 for which (1 r 0 )θl 1 L 3 (P 45 ) + r 0 φl 0 L 2 (P 45 ) 0, hence ϕ P (P 45 ) 0. For the second factor of ϕ, we let Q A, Q B, and Q C denote the clockwise labeling of the homogeneous vertices of the diagonal pentagon that do not lie on the vertical axis, and define ϕ DP (1 s)l A L C + sl B L 2. By analogy with ϕ P above, ϕ DP interpolates the points P A25, P AB14, P BC35, and P C24, duals of four of the sides of the diagonal pentagon DP. Let x A, x B, x C denote the first components of Q A, Q B, and Q C respectively and recall that P 13 [1 : 0 : 0] is the dual of the side of DP containing the origin. Since L A L C (P 13 ) x A x C > 0 and L B L 2 (P 13 ) x B x 2 < 0, there exists a unique parameter 0 < s 0 < 1 for which ϕ DP (P 13 ) 0. With this parameter choice, ϕ DP interpolates the duals of all five sides of the diagonal pentagon of P. Thus ϕ and the product ϕ P ϕ DP both interpolate the duals of the ten segments joining the vertices Q j to Q k, for distinct j, k 1, 2,..., 5. With the constraints m 1 m 2 r 0 1 r 0 m 1 m 3 θ 1 θ m 1 Q 5 + m 5 Q 1 ϕ P (P 015 ) m 3 Q 4 + m 4 Q 3 ϕ P (P 034 ) (9) we shall show that ϕ and the product ϕ P ϕ DP additionally have common gradients at four specific points. It is this observation that forces the two quartic polynomials ϕ and ϕ P ϕ DP to be proportional to one another. The gradient of Marden s function ϕ at the points P 12, P 23, P 015, and P 034 corresponds to points of interpolation for ϕ 0. Each of these interpolated points subdivides a line segment into fixed proportions according the the values of the exponents. With the exponents defined by (8) and (9), it follows that ϕ(p 12 ) m 1 Q 2 + m 2 Q 1, ϕ(p 23 ) m 2 Q 3 + m 3 Q 2, ϕ(p 015 ) m 1 Q 5 + m 5 Q 1, ϕ(p 034 ) m 3 Q 4 + m 4 Q 3 We next compute the gradient of the product ϕ P ϕ DP. In general (ϕ P ϕ DP ) ϕ P ϕ DP + ϕ P ϕ DP, but, since ϕ P is zero at the points P 12, P 23, P 015, and P 034, satisfies (ϕ P ϕ DP )(P ) ϕ P (P ) (8) January 2014] MARDEN S THEOREM FOR INSCRIBED ELLIPSES 9

10 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 10 at those points. The proof of the theorem is complete if we show that ϕ ϕ P at the four points P 12, P 23, P 015, and P 034. The proportionality is obvious at the latter two points from the constraints on m 1, m 3, m 4, and m 5 in (9). To see that ϕ and ϕ P are proportional at P 12 we note that ϕ(p 12 ) and ϕ P (P 12 ) each correspond to a point between Q 1 and Q 2. That is, comparison of the properties of the exponents in (8) with those in (5), with r r 0, imply that ϕ(p 12 ) ϕ P (P 12 ). A similar argument holds at the point P 23. Thus Marden s function ϕ is proportional to the product of ϕ P ϕ DP. The dual of each factor is an ellipse: that of ϕ P is inscribed in the pentagon, and that of ϕ DP is inscribed in its diagonal pentagon. The uniqueness of each inscribed ellipse follows because another ellipse would need to share five independent tangent conditions, forcing them to be the same. For n 6 there exist convex n-gons with no inscribed ellipses. Proof. By Brianchon s theorem, a convex hexagon contains an inscribed ellipse if and only if the diagonals intersect at a point. It is unlikely, that a generic hexagon has an inscribed ellipse. This follows from the observation that three generic lines (these can be extended to form the diagonals of a hexagon) are unlikely to be concurrent (see Figure 2). If there exists an inscribed ellipse in a convex n-gon for n 6, then it is unique. Proof. If a convex hexagon contains an inscribed ellipse, extend two suitable sides of the hexagon to form a pentagon from which the hexagon can be thought to have been cut out. The ellipse that is inscribed in the hexagon, is also inscribed in the pentagon. Since any pentagon can only inscribe a unique ellipse, this ellipse must be unique. A similar argument can be applied to n-gons for n > 6. REFERENCES 1. Maxime Bôcher. Some propositions concerning the geometric representation of imaginaries. Ann. of Math., 7(1-5):70 72, 1892/ John Clifford and Michael Lachance. A generalization of bôcher-grace theorem. Rocky Mountain J. Math., To Appear. 3. John Clifford and Michael Lachance. Quartic coincidences and the singular value decomposition. Math. Mag., To Appear. 4. Christopher Frayer, Miyeon Kwon, Christopher Schafhauser, and James A. Swenson. The geometry of cubic polynomials. Math. Mag., To Appear. 5. Pamela Gorkin and Elizabeth Skubak. Polynomials, ellipses, and matrices: two questions, one answer. Amer. Math. Monthly, 118(6): , Dan Kalman. An elementary proof of Marden s theorem. Amer. Math. Monthly, 115(4): , Ben-Zion Linfield. On the relation of the roots and poles of a rational function to the roots of its derivative. Bull. Amer. Math. Soc., 27(1):17 21, Morris Marden. Geometry of polynomials. Second edition. Mathematical Surveys, No. 3. American Mathematical Society, Providence, R.I., D. Minda and S. Phelps. Triangles, ellipses, and cubic polynomials. Amer. Math. Monthly, 115(8): , J Siebeck. Ueber eine neue analytsche behandlungweise der brennpunkte. J. Reine Angew. Math., 64(175), Joseph H. Silverman and John Tate. Rational points on elliptic curves. Undergraduate Texts in Mathematics. Springer-Verlag, New York, MAHESH AGARWAL (mkagarwa@umich.edu) received his PhD in Mathematics from the University of Michigan-Ann Arbor. He is now an Assistant Professor at the University of Michigan-Dearborn. His current 10 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 121

11 Mathematical Assoc. of America American Mathematical Monthly 121:1 October 25, :36 p.m. AgarwalAMS.tex page 11 research interests include Number Theory, Automorphic and he has recently started exploring Marden s work. He enjoys hiking, pottery, world affairs and the use of technology in higher education. Department of Mathematics and Statistics University of Michigan-Dearborn Dearborn, MI JOHN CLIFFORD (jcliff@umich.edu) received his PhD in Mathematics from Michigan State University under the direction of Joel Shapiro. He received his BS degree in mathematics from the University of Puget Sound in He is now an associate professor of mathematics at the University of Michigan-Dearborn. In his free time he enjoys watching his two boys Ben and Sam play soccer. His current research interests include operators on spaces of analytic functions and zeros of polynomials. Department of Mathematics and Statistics University of Michigan-Dearborn Dearborn, MI MICHAEL LACHANCE (malach@umich.edu) received his PhD in Mathematics from University of South Florida under the direction of E.B. Saff in He is now a professor of mathematics and chair in the Department of Mathematics & Statistics at the University of Michigan-Dearborn. Department of Mathematics and Statistics University of Michigan-Dearborn Dearborn, MI January 2014] MARDEN S THEOREM FOR INSCRIBED ELLIPSES 11