DOCSLIB.ORG

A Geometric Proof of the Siebeck-Marden Theorem

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Geometric Proof of the Siebeck-Marden Theorem A GEOMETRIC PROOF OF MARDEN'S THEOREM BENIAMIN BOGOSEL 1. Introduction and Previous Results Consider three points in the plane, whose affixes are represented by three complex numbers a; b; c. Suppose that a; b; c are not collinear, so that they form a non- degenerate triangle. We can consider the polynomial p(z) = (z − a)(z − b)(z − c), whose roots are a; b; c. It is a well known fact that given a polynomial with roots z1; :::; zn, the roots of its derivative represent points which lie in the convex hull of the points determined by z1; :::; zn. In the simplest case, where we have only three 0 roots, more can be said. In fact, if we denote the roots of p (z) by f1; f2, these two complex numbers represent two points F1;F2 which have a special property. There exists a unique ellipse which is tangent to each of the sides of the triangle determined by a; b; c, and F1;F2 are the focal points of this ellipse. This result was revisited recently in the articles of D. Kalman [2] and D. Minda and S. Phelps [3]. The above authors attribute the theorem to J. Siebeck [4] which provided a proof in 1864. Another proof was given by B^ocher in [1]. In all these articles, geometrical and algebrical tools are used in order to prove the stated result. The purpose of this note is to provide an alternative proof, which relies mostly on geometric aspects of the problem. Using the terminology adopted by Kelman in [2], we will refer to the mentioned result as Marden's Theorem. The first thing noted in all references mentioned above is the straightforward 0 observation that the roots a; b; c of the polynomial p and the roots f1; f2 of p must satisfy f + f a + b + c 1 2 = : 2 3 This algebraic relation says exactly that the centroid G of the triangle ABC coin- cides with the midpoint of the segment F1F2. The second observation is that the points f1; f2 satisfy an elementary property of an ellipse, with respect to tangent lines. Ellipses have the following interesting property property: Proposition 1.1. If A is an exterior point to an ellipse E and X1;X2 are the two focal lines of the ellipse, then the angles made by the two tangents from A to E with AX1 and, respectively, AX2 are equal. Thus, a necessary condition for F1;F2 to be the focal points of the Steiner inel- lipse is: \F1AB = \F2AC; \F1BC = \F2BA; \F1CA = \F2CB; which, in fact says that F1;F2 are isogonal conjugates in the triangle ABC. Here, the authors mentioned above, have different approaches. D. Kalman [2] makes the remark that in the case of an ellipse which is tangent to all three sides of the triangle has the property that its two focal points are isogonal conjugates. This is a direct 1 2 BENIAMIN BOGOSEL consequence of the above result. On the other hand, B^ocher [1] proves directly that F1;F2 are isogonal conjugates in the triangle determined by A; B; C using only the 0 fact that f1; f2 are the roots of p . The third aspect, underlined in [2], is the fact that performing an affine trans- formation on a; b; c, modifies f1; f2 by the same affine transformation. In this way, we are free to choose any position we want for a; b; c, as long as it is obtained using a translation and a rotation from the initial position, without affecting the result. We are now ready to present our proof, but before we do this, we present the main lines of the proofs from the cited articles. Both Kalman [2] and B^ocher [1] start their argument by choosing an ellipse which has focal points f1; f2 and is tangent to one of the sides. Then they prove that this ellipse is in fact the unique ellipse which is tangent to all sides of the triangle a; b; c at their midpoints. On the other hand, D. Minda and S. Phelps, base their proofs on the algebraic properties of complex numbers and their relation with the plane geometry. Once we have established the two basic properties satisfied by F1;F2, we can prove directly that F1;F2 are the focal points of the Steiner inellipse. First, we will present a short argument justifying that F1;F2 are indeed isogonal conjugates in the triangle ABC. Based on the third aspect presented above, whose proof can be found in [2], we can assume without loss of generality that a = 0 and b; c are oriented such that the imaginary axis is the bisectrix of the angle \BAC. It is straightforward to see that there exists θ 2 (0; π) and p; q 2 (0; 1) such that b = peiθ; c = qei(π−θ). Using 0 the fact that the roots of p satisfy f1f2 = 3(ab + bc + ca), and the choice of a = 0, we see that f1f2 = 3bc = −pq < 0. This implies, in turn, that the sum of the arguments of f1; f2 is equal to π, which is equivalent to the fact that the imaginary axis is the bisectrix of the angle F1AF2. As a straightforward consequence, we deduce that \F1AB = \F2AC. In a similar way we can deduce the analogous angle relations with respect to vertices B and C, which prove that F1;F2 are isogonal conjugates in the triangle ABC. As we have noted in the introduction, we also know that the midpoint of F1F2 is the centroid of the triangle ABC. 2. Isogonal conjugates and inscribed ellipses In the following, we will call an ellipse inscribed in a triangle, if the ellipse lies inside the closure of the triangle and is tangent to all sides of the triangle. The purpose of this section is to establish connections between inscribed ellipses, their centers and isogonal conjugate points. As we have observed above, if an ellipse is inscribed in a triangle ABC, Propo- sition 1.1 implies that its focal points are isogonal conjugates in the triangle ABC. We may ask the converse question. If we have two isogonal points F1;F2 in the triangle ABC, does there exist an ellipse with focal points F1;F2 which is inscribed in the triangle. The answer is yes, and we present a proof below. Proposition 2.1. Suppose F1;F2 are isogonal conjugates in triangle ABC. Then there exists a unique ellipse E which is inscribed in ABC and has focal points F1;F2. Proof: Consider the points X1;X2 the reflections of F1;F2 with respect to the lines AB and BC. The construction implies that BF1 = BX1; BF2 = BX2 and A GEOMETRIC PROOF OF MARDEN'S THEOREM 3 \X1BF2 = \F1BX2, which implies that X1F2 = F1X2. We denote their common value with m. If we denote A1 = F1X2\BC and C1 = X1F2\AB. The construction of X1;X2 implies that F1A1 + F2A1 = F1X2 = F2X1 = F1C1 + F2C1 = m. Furthermore, it is a classical result that A1 is the point which minimizes X 7! F1X + F2X with X 2 BC and C1 is the point which minimizes X 7! F1X + F2X with X 2 AB. Thus, the ellipse characterized by F1X + F2X = m is tangent to BC and AB in A1 and, respectively C1. A similar argument proves that this ellipse is, in fact, also tangent to AC. The unicity of this ellipse comes from the fact that m is defined as the minimum of F1X + F2X where X is on one of the sides of the triangle ABC, and this minimum is unique, and independent of the chosen side. This observation provides an interesting and simple construction of the conjugate of some point F in the triangle ABC. We consider the reflections M; N; P of F with respect to the sides AB; BC and CA of the triangle. If F 0 is the isogonal conjugate of F , then the previous arguments imply that F 0M = F 0N = F 0P , which means that F 0 is the circumcircle of the triangle MNP . Thus, constructing the isogonal conjugate is reduced to constructing the circumcenter of the triangle MNP . The center of an inscribed ellipse E is the midpoint S of the segment F1F2 determined by its focal points. We note that S is the image by F2 by a homothety of center F1 and ratio 1=2. Thus, as a direct consequence of the construction provided in the previous paragraph, we see that S is the circumcenter of the pedal 1 triangle associated to F1 A natural question is to ask what is the locus of the center of an inscribed ellipse. We are also interested to see if the center uniquely determined the inscribed ellipse. Before stating the next result, we recall some basic facts about affine transforma- tions in the plane. An affine transformation maps line to lines, preserves barycentric coordinates (midpoints, in particular) and maps ellipses to ellipses. Another impor- tant property is the fact that T1;T2 are two non-degenerate triangles in the plane, then there exists an affine transformation which maps T1 onto T2. In the same spirit, given an ellipse E, there exists an affine transformation which maps E onto a circle. Theorem 2.2. An ellipse E inscribed in the triangle ABC is uniquely determined by its center. The locus of the center of an ellipse inscribed in ABC is the interior of the median triangle of ABC2 Proof: We begin with the particular case where the ellipse E is the incircle, with center I, the incenter.
Load more

Recommended publications
  • Self-Inverse Gemini Triangles
    INTERNATIONAL JOURNAL OF GEOMETRY Vol. 9 (2020), No. 1, 25 - 39 SELF-INVERSE GEMINI TRIANGLES CLARK KIMBERLING and PETER MOSES Abstract. In the plane of a triangle ABC, every triangle center U = u : v : w (barycentric coordinates) is associated with a central triangle having A-vertex AU = −u : v + w : v + w. The triangle AU BU CU is a self-inverse Gemini triangle. Let mU (X) be the image of a point X under the collineation that maps A; B; C; G respectively onto AU ;BU ;CU ;G, where G is the centroid of ABC. Then mU (mU (X)) = X: Properties of the self-inverse mapping mU are presented, with attention to associated conics (e.g., Jerabek, Kiepert, Feuerbach, Nagel, Steiner), as well as cubics of the types pK(Y; Y ) and pK(U ∗ Y; Y ). 1. Introduction The term Gemini triangle, introduced in the Encyclopedia of Triangle Centers (ETC [7], just before X(24537)), applies to certain triangles defined by barycentric coordinates. In keeping with the meaning of gemini, these triangles tend to occur in pairs. Specifically, for a given reference triangle ABC with sidelengths a; b; c, a Germini triangle A0B0C0 is a cen- tral triangle ([6], pp. 53-56) with A-vertex given by (1.1) A0 = f(a; b; c): g(b; c; a): g(b; a; c); where f and g are center functions ([6], p. 46). Because A0B0C0 is a central triangle, the vertices B0 and C0 can be read from (1.1) as B0 = g(c; b; a): f(b; c; a): g(c; a; b) C0 = g(a; b; c): g(a; c; b): f(c; a; b): Now suppose that U = u : v : w is a triangle center, and let 0 −u v + w v + w 1 MU = @ w + u −v w + u A : u + v u + v −w Keywords and phrases Triangle, Gemini triangle, barycentric coordinates, collineation, conic, cubic, circumcircle, Jerabek, Kiepert, Feuerbach, Nagel, Steiner (2010)Mathematics Subject Classification: 51N15, 51N20 Received: 13.10.2019.
    [Show full text]
  • A Geometric Proof of the Siebeck–Marden Theorem Beniamin Bogosel
    A Geometric Proof of the Siebeck–Marden Theorem Beniamin Bogosel Abstract. The Siebeck–Marden theorem relates the roots of a third degree polynomial and the roots of its derivative in a geometrical way. A few geometric arguments imply that every inellipse for a triangle is uniquely related to a certain logarithmic potential via its focal points. This fact provides a new direct proof of a general form of the result of Siebeck and Marden. Given three noncollinear points a, b, c ∈ C, we can consider the cubic polynomial P(z) = (z − a)(z − b)(z − c), whose derivative P (z) has two roots f1, f2. The Gauss–Lucas theorem is a well-known result which states that given a polynomial Q with roots z1,...,zn, the roots of its derivative Q are in the convex hull of z1,...,zn. In the simple case where we have only three roots, there is a more precise result. The roots f1, f2 of the derivative polynomial are situated in the interior of the triangle abc and they have an interesting geometric property: f1 and f2 are the focal points of the unique ellipse that is tangent to the sides of the triangle abc at its midpoints. This ellipse is called the Steiner inellipse associated to the triangle abc. In the rest of this note, we use the term inellipse to denote an ellipse situated in a triangle that is tangent to all three of its sides. This geometric connection between the roots of P and the roots of P was first observed by Siebeck (1864) [12] and was reproved by Marden (1945) [8].
    [Show full text]
  • 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.
    [Show full text]
  • Permutation Ellipses
    Journal for Geometry and Graphics Volume 24 (2020), No. 2, 233–247 Permutation Ellipses Clark Kimberling1, Peter J. C. Moses2 1Department of Mathematics, University of Evansville, Evansville, Indiana 47722, USA [email protected] 2Engineering Division, Moparmatic Co., Astwood Bank, Nr. Redditch, Worcestershire, B96 6DT, UK [email protected] Abstract. We use homogeneous coordinates in the plane of a triangle to define a family of ellipses having the centroid of the triangle as center. The family, which includes the Steiner circumscribed and inscribed ellipses, is closed under many operations, including permutation of coordinates, complements and anticomple- ments, duality, and inversion. Key Words: barycentric coordinates, Steiner circumellipse, Steiner inellipse, com- plement and anticomplement, dual conic, inversion in ellipse, Frégier ellipse MSC 2020: 51N20, 51N15 1 Introduction In 1827, something revolutionary happened in triangle geometry. A descendant of Martin Luther named Möbius introduced barycentric coordinates. Suddenly, points and lines in the plane of a triangle ABC with sidelengths a, b, c had a new kind of representation, and traditional geometric relationships found new algebraic kinds of expression. The lengths a, b, c came to be regarded as variables or indeterminates, and a criterion for collinearity of three points became the zeroness of a determinant, as did the concurrence of three lines. More recently, algebraic concepts have been imposed on the objects of triangle geometry. You can make up a suitable function f(a, b, c) and say “Let P be the point with barycentric coordinates f(a, b, c): f(b, c, a): f(c, a, b)” and then use algebra to find geometrically notable properties of P .
    [Show full text]
  • Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse
    KoG•24–2020 R. Garcia, D. Reznik, H. Stachel, M. Helman: Steiner’s Hat: a Constant-Area Deltoid ... https://doi.org/10.31896/k.24.2 RONALDO GARCIA, DAN REZNIK Original scientific paper HELLMUTH STACHEL, MARK HELMAN Accepted 5. 10. 2020. Steiner's Hat: a Constant-Area Deltoid Associated with the Ellipse Steiner's Hat: a Constant-Area Deltoid Associ- Steinerova krivulja: deltoide konstantne povrˇsine ated with the Ellipse pridruˇzene elipsi ABSTRACT SAZETAKˇ The Negative Pedal Curve (NPC) of the Ellipse with re- Negativno noˇziˇsnakrivulja elipse s obzirom na neku nje- spect to a boundary point M is a 3-cusp closed-curve which zinu toˇcku M je zatvorena krivulja s tri ˇsiljka koja je afina is the affine image of the Steiner Deltoid. Over all M the slika Steinerove deltoide. Za sve toˇcke M na elipsi krivulje family has invariant area and displays an array of interesting dobivene familije imaju istu povrˇsinui niz zanimljivih svoj- properties. stava. Key words: curve, envelope, ellipse, pedal, evolute, deltoid, Poncelet, osculating, orthologic Kljuˇcnerijeˇci: krivulja, envelopa, elipsa, noˇziˇsnakrivulja, evoluta, deltoida, Poncelet, oskulacija, ortologija MSC2010: 51M04 51N20 65D18 1 Introduction Main Results: 0 0 Given an ellipse E with non-zero semi-axes a;b centered • The triangle T defined by the 3 cusps Pi has invariant at O, let M be a point in the plane. The Negative Pedal area over M, Figure 7. Curve (NPC) of E with respect to M is the envelope of • The triangle T defined by the pre-images Pi of the 3 lines passing through points P(t) on the boundary of E cusps has invariant area over M, Figure 7.
    [Show full text]
  • An Elementary Proof of Marden's Theorem
    An Elementary Proof of Marden’s Theorem Dan Kalman What I call Marden’s Theorem is one of my favorite results in mathematics. It es- tablishes a fantastic relation between the geometry of plane figures and the relative positions of roots of a polynomial and its derivative. Although it has a much more general statement, here is the version that I like best: Marden’s Theorem. Let p(z) be a third-degree polynomial with complex coefficients, and whose roots z1,z2, and z3 are noncollinear points in the complex plane. Let T be the triangle with vertices at z1,z2, and z3. There is a unique ellipse inscribed in T and tangent to the sides at their midpoints. The foci of this ellipse are the roots of p(z). I call this Marden’s Theorem because I first read it in M. Marden’s wonderful book [6]. But this material appeared previously in Marden’s earlier paper [5]. In both sources Marden attributes the theorem to Siebeck, citing a paper from 1864 [8]. In- deed, Marden reports appearances of various versions of the theorem in nine papers spanning the period from 1864 to 1928. Of particular interest in what follows below is an 1892 paper by Maxime Bocherˆ [1]. In his presentation Marden states the theorem in a more general form than given m1 above, corresponding to the logarithmic derivative of a product (z − z1) (z − m2 m3 z2) (z − z3) where the only restriction on the exponents m j is that they be nonzero, and with a general conic section taking the place of the ellipse.
    [Show full text]
  • Mixed Intersection of Cevians and Perspective Triangles
    15TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2012 ISGG 1–5 AUGUST, 2012, MONTREAL, CANADA MIXED INTERSECTION OF CEVIANS AND PERSPECTIVE TRIANGLES Boris ODEHNAL Dresden University of Technology, Germany ABSTRACT: Any point P in the plane of a triangle ∆ = (A,B,C) can be considered as the intersection of certain Cevians, whether P is a triangle center or not. Taking two centers, say Y and Z, we find six Cevians with a total of 11 points of intersection, among them ∆’s vertices and Y and Z. The remaining six points form two triangles ∆1 and ∆2 which are both perspective to the base triangle ∆ and to each other. The three centers of perspectivity are triangle centers of ∆ and collinear, independent of the choice of Y and Z. Furthermore any pair out of ∆, ∆1, and ∆2 has the same perspectrix which yields a closed chain of Desargues’ (103,103) configurations. Then special affine appearances of Desargues’ configurations can be obtained by a suitable choice of Y and Z. Any choice of a fixed center Y leads to exactly one center Z as the fourth point of intersection of two conic sections circumscribed to ∆ such that the perspectors P1, P2, and consequently P12 are points at infinity. For fixed center Y we obtain a quadratic transformation in ∆’s plane which transforms central lines to central conic sections. Keywords: triangle, Cevian, perspective triangles, Desargues configuration, triangle center, cross- point 1. INTRODUCTION C Many triangle centers appear as the intersection of certain Cevians. For example the incenter X1 X4 is the intersection of the interior angle bisectors h of a triangle ∆ with vertices A, B, C.
    [Show full text]
  • 104.08 Finding Triangles with Given Circum-Medial Triangle
    The Mathematical Gazette March 2020 104.08 Finding triangles with given circum-medial triangle Let △A1B1C1 be an arbitrary triangle and P an arbitrary point not on its circumcircle Σ. The ‘other’ intersection points of the ‘cevians’ A1PB,,1P C1P with Σ, the points AB,,,C make the so-called circumcevian triangle of △A1B1C1 with respect to PP. If is the centroid of △A1B1C1 then △ABC is called * the circum-medial triangle of △A1B1C1 (see Figure 1: P = centroid G1). C1 A B1 B G1 A1 C FIGURE 1: How to reverse the process of constructing the circum-medial triangle? A recent paper [1] was concerned with iterating the construction of the circum-medial triangle (it turned out that the shape of the triangles converges to equilaterality). Here, we reverse the process. Specifically, given △ABC we look for △A1B1C1 such that △ABC is the circum-medial triangle of △A1B1C1. It turns out that there are in general two solutions, namely, the circumcevian triangles of △ABC with respect to the foci of the Steiner inellipse of △ABC. Theorem 1: Given △ABCGand a point 1 not on its circumcircle Σ, let △A1B1C1 be its circumcevian triangle with respect to G1. Then G1 is the centroid of △A1B1C1 if, and only if, it is a focus of the Steiner inellipse of △ABC. Remark: Since an ellipse has two foci, this yields the two mentioned solutions (see Figure 2), except in the case where △ABC is equilateral, when * Unfortunately, not every language has such short and precise terms for the corresponding triangles. For instance in German there is no such term, one would have to describe the whole process with many words.
    [Show full text]
  • Dao Thanh Oai, Generalizations of Some Famous Classical Euclidean
    International Journal of Computer Discovered Mathematics (IJCDM) ISSN 2367-7775 c IJCDM September 2016, Volume 1, No.3, pp.13-20. Received 15 July 2016. Published on-line 22 July 2016 web: http://www.journal-1.eu/ c The Author(s) This article is published with open access1. Generalizations of some famous classical Euclidean geometry theorems Dao Thanh Oai Cao Mai Doai, Quang Trung, Kien Xuong, Thai Binh, Vietnam e-mail: [email protected] Abstract. In this note, we introduce generalizations of some famous classical Euclidean geometry theorems. We use problems in order to state the theorems. Keywords. Euclidean geometry, Triangle geometry. Mathematics Subject Classification (2010). 51-04, 68T01, 68T99. Problem 1 ([1],[2], A generalization the Lester circle theorem associated with the Neuberg cubic). Let P be a point on the Neuberg cubic. Let PA be the reflection of P in line BC, and define PB and PC cyclically. It is known that the lines APA, BPB, CPC concur. Let Q(P ) be the point of concurrence. Then two Fermat points, P , Q(P ) lie on a circle. Remark: Let P is the circumcenter, it is well-know that Q(P ) is the Nine point center, the conjucture becomes Lester circle theorem, you can see the Lester circle theorem in([3],[4]). Problem 2 ([5],[6], A generalization of the Parry circle theorem associated with two isogonal conjugate points). Let a rectangular circumhyperbola of ABC, let L be the isogonal conjugate line of the hyperbola. The tangent line to the hyperbola at X(4) meets L at point K.
    [Show full text]
  • An Area Inequality for Ellipses Inscribed in Quadrilaterals
    Journal of Mathematical Inequalities Volume 4, Number 3 (2010), 431–443 AN AREA INEQUALITY FOR ELLIPSES INSCRIBED IN QUADRILATERALS ALAN HORWITZ (Communicated by S. S. Gomis) Area(E) Abstract. If E is any ellipse inscribed in a convex quadrilateral, –D, then we prove that Area(–D) π , and equality holds if and only if –D is a parallelogram and E is tangent to the sides of –D 4 at the midpoints. We also prove 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. This does not hold in general for convex quadrilaterals. 1. Introduction It is well known (see [1], [4], or [5]) that there is a unique ellipse inscribed 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 √ , (1) Area(T) 3 3 with equality if and only if E is the midpoint ellipse. In [5] the authors also discuss a connection between the Steiner ellipse and the orthogonal least squares line for the vertices of T . DEFINITION 1. The line, £, is called a line of best fitforn given points z1,...,zn n 2 in C,if£ minimizes ∑ d (z j,l) among all lines l in the plane. Here d (z j,l) denotes j=1 the distance(Euclidean) from z j to l .
    [Show full text]
  • Perspective Fields-Part2 Size
    Part 2 – Algebraic Approach and Theory © Chris van Tienhoven [email protected] There is a most special application for Perspective Fields. Because points are generated structurally in Perspective Fields this also can be seen in the formulas of the points. All points in the field we described earlier (defined by X13, X2, X15, X6) have the same algebraic structure in the formulas of their barycentric coordinates: f (a,b,c) = i a2 SA + j SB SC + k a2∆ Where: ∆ = Area of Reference Triangle a,b,c are lenghts of sides Reference Triangle SA = (-a2 + b2 + c2 ) / 2 SB = ( a2 - b2 + c2 ) / 2 SC = ( a2 + b2 - c2 ) / 2 i, j, k are variables uniquely defining the points in a Perspective Field So every point in the field can be defined as a triple (i : j : k). Because only the ratios of variables i, j, k are of importance the triple can be considered as alternate homogeneous coordinates for fieldpoints. The importance of this is that a fieldpoint now is: independant of sides and angles of the referencetriangle and uniquely defined just by 3 numbers. Look at next picture to see what structure comes out. 2 Homogeneous Coordinates (i : j : k) with respect to barycentric coordinates f (a,b,c) = i a2 SA + j SB SC + k a2∆ Note that for all Fieldpoints: . on X4-X6-Line 1st coordinate = 0 . on BrocardAxis 2nd coordinate = 0 . on Euler Line 3rd coordinate = 0 Note that this formula-structure is only valid for Fieldpoints. Non-fieldpoints will have a different formula-structure. 3 Point of departure: .
    [Show full text]
  • The Siebeck–Marden–Northshield Theorem and the Real Roots
    The Siebeck–Marden–Northshield Theorem and the Real Roots of the Symbolic Cubic Equation Emil M. Prodanov School of Mathematical Sciences, Technological University Dublin, Park House, Grangegorman, 191 North Circular Road, Dublin D07 EWV4, Ireland, e-mail: [email protected] Abstract The isolation intervals of the real roots of the symbolic monic cubic polynomial x3 ax2 bx c are determined, in terms of the coefficients of the polynomial, by` solving` the Siebeck–Marden–Northshield` triangle — the equilateral triangle that projects onto the three real roots of the cubic polynomial and whose inscribed circle projects onto an interval with endpoints equal to stationary points of the polynomial. Mathematics Subject Classification Codes (2020): 26C10, 12D10, 11D25. Keywords: Polynomials; Cubic equation; Siebeck–Marden–Northshield theorem; Roots; Isolation intervals; Root bounds. arXiv:2107.01847v1 [math.GM] 5 Jul 2021 1 1 Introduction The elegant theorem of Siebeck and Marden (often referred to as Marden’s theorem) [1]–[5] relates geometrically the complex non-collinear roots of a cubic polynomial with complex coefficients to a triangle whose vertices project onto them, on one hand, and, on the other, the critical points of the polynomial to the projections of the foci of the inellipse of this triangle. This ellipse is unique and is called Steiner inellipse [6]. It is inscribed in the triangle in such way that it is tangent to the sides of the triangle at their midpoints. The real version of the Siebeck–Marden Theorem, as given by Northshield [7], states that the three real roots (not all of which are equal) of a cubic polynomial are projections of the vertices of some equilateral triangle in the plane.
    [Show full text]