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 Bôcher 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 inellipse 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, Bôcher [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 transformation 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 Bôcher [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 ∈ (0, π) and p, q ∈ (0, ∞) 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 ∈ BC and C1 is the point which minimizes X 7→ F1X + F2X with X ∈ 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 transformations 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. Suppose E0 is another inscribed ellipse, with center I, and denote F1,F2 its focal points. We know that F1,F2 are isogonal conjugates in ABC and the midpoint of F1F2 is I, the center of the ellipse. Thus, if F1 6= F2 then AI is at the same time median and bisectrix in triangle AF1F2. This implies that AI ⊥ F1F2. A similar argument proves that BI ⊥ F1F2 and CI ⊥ F1F2. Thus A, B, C all lie on a line perpendicular to F1F2 in I, which contradicts the fact that ABC is not degenerate. The assumption F1 6= F2 leads to a contradiction, and 0 0 therefore we must have F1 = F2, which means that E is a circle and E = E. Consider now the general case. Suppose that the ellipses E, E0 have the same center and they are both inscribed in ABC. Consider an affine mapping h which

1The triangle determined by the projections of a point on the sides of a triangle is called pedal triangle. 2The median triangle is determined by the midpoints of the sides of ABC 4 BENIAMIN BOGOSEL maps E to a circle. Since h maps ellipses to ellipses and preserves midpoints, the image of our configuration by h is a triangle where h(E) is the incenter and h(E0) is an inscribed ellipse. This case was treated in the previous paragraph and we must have h(E) = h(E0). Thus E = E0. To find the locus of the center of an inscribed ellipse, it is enough to see which barycentric coordinates are admissible for the incircle of a triangle. Indeed, barycentric coordinates are preserved under affine transformations. The barycentric coordinates of the center of an inscribed ellipse with respect to A, B, C are the same as the barycentric coordinates of the incenter of the triangle h(A), h(B), h(C). Like before, h is the affine transformation which transforms the ellipse into a circle. Con- versely, if the barycentric coordinates of the intcenter I with respect to A0B0C0 are x, y, z, then we consider the affine transformation which maps the triangle A0B0C0 onto the triangle ABC. The circle will be transformed into an inscribed ellipse, with center having barycentric coordinates x, y, z. The barycentric coordinates of the incenter have the form u v w x = , y = , z = , u + v + w u + v + w u + v + w where u, v, w are the sides of the triangle A0B0C0. Thus x, y, z are characterized like the sides of a non degenerate triangle. Since Area(IBC) Area(ICA) Area(IAB) x = , y = , z = , Area(ABC) Area(ABC) Area(ABC) we can see that the inequalities x < y + z, y < z + x, z < x + y are satisfied if and only if I is in the interior of the median triangle. Thus, the locus of the center of an inscribed ellipse is the interior of the median triangle. Using Proposition 2.1 and Theorem 2.2 we can immediately prove Marden’s Theorem.

Corollary 2.3. If a, b, c are roots of a third degree polynomial p and f1, f2 are 0 the roots of p , then the points with aﬃxes f1, f2 are the focal points of the Steiner inellipse corresponding to the triangle deﬁned by a, b, c.

Proof: We know that F1,F2 are isogonal conjugates, so there exists a unique ellipse E inscribed in the triangle ABC with focal points F1,F2. Since the midpoint of F1F2 is the centroid of ABC, the center of the ellipse E is also the centroid of ABC. On the other hand, the centroid of the Steiner inellipse is the centroid of ABC. Since by Theorem 2.2, the center of an inscribed ellipse uniquely determines the ellipse, we can conclude that E is the Steiner inellipse. References [1] Maxime Bˆocher. Some propositions concerning the geometric representation of imaginaries. Ann. of Math., 7(1-5):70–72, 1892/93. [2] Dan Kalman. An elementary proof of Marden’s theorem. Amer. Math. Monthly, 115(4):330– 338, 2008. [3] D. Minda and S. Phelps. Triangles, ellipses, and cubic polynomials. Amer. Math. Monthly, 115(8):679–689, 2008. [4] J. Siebeck. Ueber eine neue analytische behandlungweise der brennpunkte. J. Reine Angew. Math. 64, 1864.