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An Area Inequality for Ellipses Inscribed in Quadrilaterals

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An Area Inequality for Ellipses Inscribed in Quadrilaterals 1 Introduction It is well known(see [CH], [K], or [MP]) that there is a unique ellipse in- scribed in a given triangle, T , tangent to the sides of T at their respective midpoints. This is often called the midpoint or Steiner inellipse, and it can be characterized as the inscribed ellipse having maximum area. In addition, one has the following inequality. If E denotes any ellipse inscribed in T , then Area (E) π , (trineq) Area (T ) ≤ 3√3 with equality if and only if E is the midpoint ellipse. In [MP] the authors also discuss a connection between the Steiner ellipse and the orthogonal least squares line for the vertices of T . Definition: The line, £, is called a line of best fit for n given points z1, ..., zn n 2 in C, if £ minimizes d (zj ,l) among all lines l in the plane. Here d (zj,l) j=1 denotes the distance(Euclidean)P from zj to l. In [MP] the authors prove the following results, some of which are also proven in 1 n Theorem A: Suppose zj are points in C,g = zj is the centroid, and n j=1 n n 2 2 2 P Z = (zj g) = zj ng . j=1 − j=1 − (a)P If Z = 0, thenP every line through g is a line of best fit for the points z1, ..., zn. (b) If Z = 0, then the line, £, thru g that is parallel to the vector from 6 (0, 0) to √Z is the unique line of best fit for z1, ..., zn. Theorem B: If z1,z2, and z3 are the vertices of a nonequilateral triangle, 3 2 T , and £ minimizes d (zk,l) among all lines l in the plane, then the major k=1 axis of the Steiner ellipseP lies on £. A proof of Theorem B goes all the way back to Coolidge in 1913(see [MP]). The purpose of this paper is to attempt to generalize (trineq) and Theorem B to ellipses inscribed in convex quadrilaterals. Many of the results in this arXiv:0907.1647v1 [math.CA] 9 Jul 2009 paper use results from two earlier papers of the author about ellipses inscribed in quadrilaterals. In particular, in [H1] we proved the following results. Theorem C: Let D be a convex quadrilateral in the xy plane and let M1 and M2 be the midpoints of the diagonals of D. Let Z be the open line segment connecting M1 and M2. If(h, k) Z then there is a unique ellipse with center (h, k) inscribed in D. ∈ Theorem D: Let D be a convex quadrilateral in the xy plane. Then there is a unique ellipse of maximal area inscribed in D. The following general result about ellipses is essentially what appears in [W], except that the cases with A = B were added by the author. 1 Lemma E: Let E be an ellipse with equation Ax2 + By2 +2Cxy + Dx + Ey + F = 0, and let φ denote the counterclockwise angle of rotation from the line thru the center parallel to the x axis to the major axis of E. Then 0 if C = 0 and A<B π if C = 0 and A>B 2 1 −1 A B cot − if C = 0 and A<B 2 2C 6 φ = π 1 −1 A B + cot − if C = 0 and A>B 2 2 2C 6 π if C < 0 and A = B 4 3π if C > 0 and A = B 4 In [H2], we also derived the following results about ellipses inscribed in parallelograms. Lemma F: Let Z be the rectangle with vertices (0, 0), (l, 0), (0, k),and (l, k), where l,k > 0. (A) The general equation of an ellipse, E, inscribed in Z is given by k2x2 + l2y2 2l (k 2v) xy 2lkvx 2l2vy + l2v2 =0, 0 < v < k. − − − − (B) The corresponding points of tangency of Ψ are lv l , 0 , (0, v), (k v) , k , and (l, k v). k k − − (C) If a and b denote the lengths of the semi–major and semi–minor axes, respectively, of E, then 2 2 2l (k v) v a = − and k2 + l2 (k2 + l2)2 16l2 (k v) v − 2 − − 2 2l (k v) v b = p − . k2 + l2 + (k2 + l2)2 16l2 (k v) v − − Proposition G: Let P be thep parallelogram with vertices O = (0, 0), P = (l, 0),Q = (d, k), and R = (d + l, k), where l,k > 0, d 0. The general equation of an ellipse, E, inscribed in P is given by ≥ 3 2 2 2 k x + k(d + l) 4dlv y 2k (k(d + l) 2lv) xy 2 − − 2 2 − 2k lvx +2klv (d l) y + kl v =0, 0 < v < k. − − Of course, any two triangles are affine equivalent, while the same is not true of quadrilaterals–thus it is not surprising that not all of the results about ellipse inscribed in triangles extend nicely to quadrilaterals. For example, there is not necessarily an ellipse inscribed in a given quadrilateral, D, which is tangent to 2 the sides of D at their respective midpoints. There is such an ellipse, which we call the midpoint ellipse, when D is a parallelogram(see Proposition 2.2). We are able to prove an inequality(see Theorem 2.1), similar to (trineq), which holds for all convex quadrilaterals. If E is any ellipse inscribed in a quadrilateral, D, Area (E) π then , and equality holds if and only if D is a parallelogram and Area(D) ≤ 4 E is the midpoint ellipse. Not suprisingly, Theorem B also does not extend in general to ellipses in- scribed in convex quadrilaterals. However, such a characterization does hold again when D is a parallelogram. We prove in Theorem 3.1 that the foci of the unique ellipse of maximal area inscribed in a parallelogram, D, lie on the orthogonal least squares line for the vertices of D. It is also well known that if p(z) is a cubic polynomial with roots at the vertices of a triangle, then the roots of p′(z) are the foci of the Steiner inellipse. A proof of this fact goes all the way back to Siebeck in 1864(see [MP]) and is known in the literature as Marden’s Theorem. Now it is easy to show that the orthogonal least squares line, £, from Theorem A is identical to the line through the roots of p′(z). Thus Marden’s Theorem implies Theorem B and hence is a stronger statement than Theorem B. There is an obvious way to try to generalize such a result to convex quadrilaterals, D. If p(z) is a quartic polynomial with roots at the vertices of D, must the foci of the unique ellipse of maximal area inscribed in D equal the roots of p′′(z) ? We give an example in section 3 that shows that such a stronger statement does not hold for parallelograms, or even for rectangles. 2 An Area Inequality Theorem 2.1: Let E be any ellipse inscribed in a convex quadrilateral, D. Area (E) π Then , and equality holds if and only if D is a parallelogram and Area(D) ≤ 4 E is tangent to the sides of D at the midpoints. Before proving Theorem 2.1, we need the following lemma. Lemma 2.2: Suppose that s and t are positive real numbers with s + t> 1 and s =1 = t. Let 6 16 1 h = st + t 2s 1+ (t 1)2 + s2(t2 t + 1) s(t2 3t + 2) . a 6 t 1 − − − − − − − 1 1 Then h I = the open interval withp endpoints and s. a ∈ 2 2 Proof of Lemma 1: 1 st 2 (s + t 1)+ (t 1)2 + s2(t2 t + 1) s(t2 3t + 2) ha = − − − − − − , and − 2 p 6 (t 1) − (1) 1 1 h s = (t 1)2 + s2(t2 t + 1) s(t2 3t + 2) (2st (s + t 1)) . a−2 6 (t 1) − − − − − − − − p (2) 3 There are four cases to consider: s,t > 1, s< 1 <t, t< 1 <s, and s,t < 1. We prove the first two cases, the proof of the other two being similar. 1 1 Case 1: s,t > 1, which implies that I = , s 2 2 1 By (1), h > 0 a − 2 ⇐⇒ (t 1)2 + s2(t2 t + 1) s(t2 3t + 2) > 2 (s + t 1) st, which always holds since− − − − − − p 2 2 2 2 2 (t 1) +s (t t+1) s(t 3t+2) (2 (s + t 1) st) = 3 (t 1) (s 1) (s + t 1) > 0 − − − − − − − − − − 1 By (2), h s< 0 a − 2 ⇐⇒ (t 1)2 + s2(t2 t + 1) s(t2 3t + 2) < 2st (s + t 1), which always holds since− − − − − − p 2 2 2 2 2 (2st (s + t 1)) (t 1) + s (t t + 1) s(t 3t + 2) =3st (t 1) (s 1) > 0 − − − − − − − − − 1 1 Case 2: s< 1 <t, which implies that I = s, 2 2 1 By (1), h < 0 a − 2 ⇐⇒ (t 1)2 + s2(t2 t + 1) s(t2 3t + 2) < 2 (s + t 1) st, which always holds since− − − − − − p 2 2 2 2 2 (2 (s + t 1) st) (t 1) + s (t t + 1) s(t 3t + 2) = 3 (t 1)(1 s) (s + t 1) > 0 − − − − − − − − − − 1 By (2), h s> 0 a − 2 ⇐⇒ (t 1)2 + s2(t2 t + 1) s(t2 3t + 2) > 2st (s + t 1), which always holds since− − − − − − p 2 2 2 2 2 (t 1) + s (t t + 1) s(t 3t + 2) (2st (s + t 1)) =3st (t 1)(1 s) > 0 − − − − − − − − − Proof of Theorem 2.1: We shall prove Theorem 2.1 when D is not a trapezoid, though it certainly holds in that case as well.
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