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 nondegenerate 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
roots, more can be said. In fact, if we denote the roots of p0 (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
observation that the roots a, b, c of the polynomial p and the roots f1 , f2 of p0 must
satisfy
a+b+c
f1 + f2
=
.
2
3
This algebraic relation says exactly that the centroid G of the triangle ABC coincides with the midpoint of the segment F1 F2 .
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:
∠F1 AB = ∠F2 AC, ∠F1 BC = ∠F2 BA, ∠F1 CA = ∠F2 CB,
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
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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
fact that f1 , f2 are the roots of p0 .
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
the fact that the roots of p0 satisfy f1 f2 = 3(ab + bc + ca), and the choice of a = 0,
we see that f1 f2 = 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 F1 AF2 .
As a straightforward consequence, we deduce that ∠F1 AB = ∠F2 AC. 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 F1 F2 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, Proposition 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
∠X1 BF2 = ∠F1 BX2 , which implies that X1 F2 = F1 X2 . We denote their common
value with m. If we denote A1 = F1 X2 ∩BC and C1 = X1 F2 ∩AB. The construction
of X1 , X2 implies that F1 A1 + F2 A1 = F1 X2 = F2 X1 = F1 C1 + F2 C1 = m.
Furthermore, it is a classical result that A1 is the point which minimizes X 7→
F1 X + F2 X with X ∈ BC and C1 is the point which minimizes X 7→ F1 X + F2 X
with X ∈ AB.
Thus, the ellipse characterized by F1 X + F2 X = 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 F1 X + F2 X 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 0 M = F 0 N = F 0 P , which means
that F 0 is the circumcircle of the triangle M N P . Thus, constructing the isogonal
conjugate is reduced to constructing the circumcenter of the triangle M N P .
The center of an inscribed ellipse E is the midpoint S of the segment F1 F2
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
triangle associated to F1 1 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 important 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 ABC 2
Proof: We begin with the particular case where the ellipse E is the incircle, with
center I, the incenter. Suppose E 0 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 F1 F2 is I, the center of the ellipse. Thus, if F1 6= F2 then AI
is at the same time median and bisectrix in triangle AF1 F2 . This implies that
AI ⊥ F1 F2 . A similar argument proves that BI ⊥ F1 F2 and CI ⊥ F1 F2 . Thus
A, B, C all lie on a line perpendicular to F1 F2 in I, which contradicts the fact that
ABC is not degenerate. The assumption F1 6= F2 leads to a contradiction, and
therefore we must have F1 = F2 , which means that E 0 is a circle and E 0 = E.
Consider now the general case. Suppose that the ellipses E, E 0 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
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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(E 0 ) is
an inscribed ellipse. This case was treated in the previous paragraph and we must
have h(E) = h(E 0 ). Thus E = E 0 .
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. Conversely, if the barycentric coordinates of the intcenter I with respect to A0 B 0 C 0 are
x, y, z, then we consider the affine transformation which maps the triangle A0 B 0 C 0
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
v
w
u
, y=
, z=
,
x=
u+v+w
u+v+w
u+v+w
where u, v, w are the sides of the triangle A0 B 0 C 0 . Thus x, y, z are characterized
like the sides of a non degenerate triangle. Since
Area(ICA)
Area(IAB)
Area(IBC)
,y =
,z =
,
x=
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
the roots of p0 , 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 F1 F2 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 Bôcher. 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.