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5 ReferencesTriangles, ellipses, and cubic polynomials
Abstract
There are a number of very interesting geometric connections between the roots of a cubic polynomial and the roots of the derivative. Precisely, suppose p(z) is a monic cubic polynomial whose roots form the vertices of a triangle, Δ, in the complex plane C. It is elementary that the centroid of Δ is the midpoint, or centroid, of the roots of p′(z). The Gauss-Lucas Theorem implies that the roots of p′(z) lie inside Δ. Siebeck's Theorem (1864) asserts that the roots of p′(z) are the foci of the Steiner inellipse for Δ. The Steiner inellipse for a triangle is the unique inscribed ellipse that is tangent to the sides at their midpoints. A result of Coolidge (1913) asserts that the line of best fit for the vertices of the triangle is the line through the roots of p′(z). These known geometric connections are established using elementary properties of complex numbers and affine transformations.
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Triangles, Ellipses, and Cubic Polynomials
D. Minda and S. Phelps
1. INTRODUCTION. Discussions that led to this paper began during an electronic
version of the Secondary School Teachers Program, a part of the Park City Mathemat-
ics Institute (PCMI), offered at the University of Cincinnati in the summer of 2006 for
local high school mathematics teachers. The authors were working with the teachers
on daily problem sets provided by PCMI; complex numbers and polynomials were a
theme throughout the problems. The second author raised the question of geometric
connections between the triangle whose vertices are the roots of a cubic polynomial
and the roots of the derivative of the polynomial. He had noticed some relationships,
such as that the roots of the derivative were inside the triangle, by performing exper-
iments using Geometers Sketchpad. The second author was able to justify some of
the observed relations. We continued to investigate the matter for the three weeks of
the program. Roughly speaking, we followed the trail of results described in the next
section. The fundamental connection is that if z1,z2,z3are the vertices of a triangle
and p(z)=(z−z1)(z−z2)(z−z3), then the roots of p(z)are the foci of the Steiner
inellipse for the triangle. The Steiner inellipse is the unique ellipse that is inscribed in
the triangle and tangent to the sides at their midpoints. The inellipse degenerates to a
circle precisely when the triangle is equilateral, and this occurs if and only if p(z)has
a repeated root. The Steiner inellipse for a triangle has the largest area among all el-
lipses contained in the triangle. A related result is that for a nonequilateral triangle the
unique line of best fit for the vertices of the triangle is the line through the foci of the
inellipse. These attractive results deserve to be better known; we offer an elementary
approach using basic complex analysis and simple affine geometry.
2. MAIN RESULTS. There are a number of interesting connections between the
roots of a polynomial p(z)=anzn+an−1zn−1+···+a1z+a0,wherean= 0, and
the roots of the derivative p(z)=nanzn−1+(n−1)an−1zn−2+···+a1. We recall
some of these.
Perhaps the best known result of this type is the Gauss-Lucas theorem, which asserts
that the zeros of the derivative lie in the convex hull of the zeros of the polynomial [1,
p. 29]. Another result of this type is that the centroid of the zeros of a polynomial equals
the centroid of the roots of the derivative. This latter property is easily verified from a
connection between the roots of a polynomial and the coefficients. If zj,1≤j≤n,
are the roots of p(z),thenan−1/an=−(z1+···+zn). Similarly, if zj,1 ≤j≤n−1,
are the roots of the derivative, then
(n−1)an−1
nan=−(z
1+···+z
n−1).
Therefore,
1
n(z1+···+zn)=1
n−1(z
1+···+z
n−1). (2.1)
In particular, for a quadratic polynomial, the root of the derivative is the midpoint of the
segment determined by the roots of the quadratic; for a real quadratic polynomial with
October 2008] TRIANGLES, ELLIPSES, AND CUBIC POLYNOMIALS 679
distinct real roots this is the familiar result that the vertex of a parabola lies midway
between the points at which the parabola crosses the x-axis.
The focus of this paper is on connections between the roots of a cubic polynomial
and those of the quadratic derivative. We begin by establishing some elementary con-
nections between these roots. We consider only the generic case in which the roots of
the cubic are distinct and noncollinear, say the roots are z1,z2,z3. We may assume the
cubic is monic, p(z)=(z−z1)(z−z2)(z−z3), since division by a nonzero constant
does not change the roots of a polynomial or its derivative. Note that
p(z)=z3−(z1+z2+z3)z2+(z1z2+z2z3+z1z3)z−z1z2z3(2.2)
and
p(z)=3z2−2(z1+z2+z3)z+(z1z2+z2z3+z1z3). (2.3)
It is convenient to let g=1
3(z1+z2+z3), which is the centroid of the triangle z1z2z3
determined by the roots of the cubic. The roots of p(z)are
g±g2−1
3(z1z2+z2z3+z1z3);(2.4)
the roots are symmetric about the centroid. By the Gauss-Lucas theorem both roots of
p(z)lie inside or on z1z2z3. In fact, the roots of p(z)lie inside the triangle. To see
this, suppose zis a root of p(z). Observe that z= zj,j=1,2,3, since p(z)has no
repeated roots. Then
0=p(z)
p(z)=1
z−¯z1+1
z−¯z2+1
z−¯z3
=z−z1
|z−z1|2+z−z2
|z−z2|2+z−z3
|z−z3|2.
This gives z=α1z1+α2z2+α3z3,where
αk=|z−zk|−2
3
j=1|z−zj|−2>0
and α1+α2+α3=1. Since zis a proper convex combination of z1,z2,z3, it lies
inside z1z2z3. Next, we show that p(z)has a single repeated root (at the centroid) if
and only if z1z2z3is equilateral. From (2.4), p(z)has a repeated root if and only if
3g2=z1z2+z2z3+z1z3,(2.5)
or equivalently,
z2
1+z2
2+z2
3=z1z2+z2z3+z1z3.(2.6)
It is a standard exercise in complex analysis ([1,p.15]and[5, p. 32]) to show that the
identity (2.6) holds if and only if z1,z2,z3are the vertices of an equilateral triangle.
Here is a simple argument to establish this result. By making use of (2.5) we “complete
the cube” and obtain
p(z)=z3−3gz2+3g2z−z1z2z3=(z−g)3−z1z2z3+g3.
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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 115
If ζdenotes a cube root of z1z2z3−g3, then the roots of p(z)are g+ζ,g+ωζ ,
g+ω2ζ,whereω=exp(2πi/3). These points are the vertices of an equilateral tri-
angle.
A more sophisticated connection between the roots of a cubic and those of the
derivative follows from a lovely geometric result of Steiner. The incircle of a triangle
is tangent to the sides and the three points of tangency are the midpoints of the sides if
and only if the triangle is equilateral. Steiner showed that it is always possible to find
an ellipse inscribed in a triangle that is tangent at the midpoints of the sides.
Theorem 2.1 (Steiner). Given any triangle there is a unique ellipse inscribed in the
triangle that passes through the midpoints of the sides of the triangle and is tangent to
the sides of the triangle at these midpoints. If z1,z2,z3are the vertices of the triangle,
then the foci of this ellipse are
g±g2−1
3(z1z2+z2z3+z1z3), (2.7)
where g =1
3(z1+z2+z3)is the centroid.
The ellipse that is inscribed in z1z2z3and tangent at the midpoints of the sides
is called the Steiner inellipse. Note that the center of the Steiner inellipse is at the
centroid of the triangle. The inellipse degenerates to a circle if and only if z1z2z3is
equilateral. From (2.4) and (2.7) we immediately obtain the following result.
Corollary 2.2 (Siebeck). Suppose z1,z2,z3are noncollinear points in Cand p(z)=
(z−z1)(z−z2)(z−z3). Then the roots of p(z)are the foci of the Steiner inellipse.
This corollary is due to Siebeck [6]. His result has been reproved and extended by
a number of authors; see [4, pp. 9–13] for details.
Another appealing connection was noted by Coolidge [2]. For a set of npoints zj
in C,1≤j≤n, consider the line(s) that best fit these points in the following sense.
Given a line ,letdist(zj,)denote the distance from zjto . A line for which
D=
n
j=1
dist2(zj,) (2.8)
is minimized is called a line of best fit for the points zj,1≤j≤n.Forn=2the
unique line of best fit is the line through the two points. Note that the sum of the
squares of orthogonal distances is being minimized, not the sum of the squares of
the vertical separations used in determining a regression line. Also, if fis any Eu-
clidean similarity and is a line of best fit for z1,... ,zn,then f() is a line of best fit
for f(z1),... , f(zn), a property not shared by the regression line.
Theorem 2.3. Suppose z j,1≤j≤n, are points in C,g=1
nn
j=1zjis the centroid,
and Z =n
j=1(zj−g)2=n
j=1z2
j−ng2.
(a) If Z =0, then every line through g is a line of best fit for the points z1,... ,zn.
(b) If Z = 0, then the line through g that is parallel to the vector from 0to √Zis
the unique line of best fit for z1,... ,zn.
October 2008] TRIANGLES, ELLIPSES, AND CUBIC POLYNOMIALS 681
For three noncollinear points z1,z2,z3,
Z=
3
j=1
z2
j−3g2=6g2−1
3(z1z2+z2z3+z1z3).
From (2.5) Z=0 if and only if z1,z2,z3are the vertices of an equilateral triangle.
Thus, if zj,1≤j≤3, are the vertices of an equilateral triangle, then every line
through the centroid is a line of best fit. If z1z2z3is not equilateral, then the unique
line of best fit is the line through gthat is parallel to g2−1
3(z1z2+z2z3+z1z3).
Corollary 2.4 (Coolidge). Suppose z1z2z3is nonequilateral. The line through the
foci of the Steiner inellipse for z1z2z3is the line of best fit for z1,z2,z3. Equivalently,
if p(z)=(z−z1)(z−z2)(z−z3), then the line of best fit is the line through the roots
of p(z).
Proof. The line of best fit is the line through the centroid gthat is parallel to
g2−1
3(z1z2+z2z3+z1z3). From (2.4) this vector is parallel to the vector join-
ing the roots of p(z), so the line of best fit is the line through the roots of p(z).
Let p(z)be a given cubic polynomial with noncollinear roots, z1,z2,z3.Forany
λ∈C,let pλ(z)=p(z)+λ. This specifies a one-parameter family of cubic poly-
nomials and most of the pλ(z)will have noncollinear roots, say z1(λ), z2(λ), z3(λ).
Since p
λ(z)=p(z), the inellipses for the triangles z1(λ)z2(λ)z3(λ) are all confocal
and the vertices of the triangles have the same line of best fit, independent of λ.Also,
all of the triangles z1(λ)z2(λ)z3(λ) have the same centroid. This is a very remarkable
situation.
The remainder of the paper is devoted to establishing Theorems 2.1 and 2.3.
3. LINEAR AND AFFINE TRANSFORMATIONS. Our proof of Steiner’s theo-
rem makes use of affine transformations, so we recall the necessary facts about linear
and affine transformations of the Cartesian plane R2. We regard the plane as the set C
of complex numbers and write transformations in terms of complex numbers by iden-
tifying the point x
yof R2with the complex number z=x+iy. In terms of complex
numbers, a real linear transformation fhas the form
f(z)=Az +B¯z,
where A=a1+ia2and B=b1+ib2.IfA= 0andB=0, then f(z)=Az is an
orientation preserving Euclidean similarity. If A=0and B= 0, then f(z)=B¯zis
an orientation reversing Euclidean similarity. The matrix representation of fis fx
y=
Mx
y,where
M=a1+b1−a2+b2
a2+b2a1−b1.
The linear transformation fis nonsingular if and only if 0 = det M=|A|2−|B|2.
Thus, fis bijective if and only if |A| =|B|.If fis nonsingular, then the image of any
circle about the origin is an ellipse or a circle with center at the origin. It is convenient
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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 115
to let T={z:|z|=1}denote the unit circle with center at the origin and rTthe circle
about the origin with radius r>0. Conversely, given an ellipse Ewith center at the
origin and any circle rT, there is a nonsingular linear transformation that maps the
circle rTonto the ellipse E.
In complex notation, an affine transformation has the form f(z)=Az +B¯z+C,
where Cis a complex number; an affine transformation is a linear transformation fol-
lowed by a translation. The defining property of an affine transformation is that for all
z,w ∈Cand any t∈R,
f(1−t)z+tw=(1−t)f(z)+tf(w). (3.1)
Affine geometry is the study of properties that are invariant under affine transforma-
tions. Affine transformations do not preserve length or angle. On the other hand, an
affine transformation maps lines to lines, parallel lines to parallel lines, midpoints to
midpoints, and centroids to centroids. That midpoints are preserved follows from (3.1)
with t=1/2. Affine transformations also preserve tangency. Any pair of triangles are
affinely equivalent.
Theorem 3.1. If f (z)=Az +B¯z+C is a bijective affine transformation, then the
image of the circle rTis the ellipse with foci C ±2r√AB, semi-major axis with length
(|A|+|B|)r, and semi-minor axis with length |A|−|B|r.
Proof. It suffices to consider the case in which C=0, f(z)=Az +B¯zis a bijective
linear transformation, and r=1since f(rz)=rf(z).If A=0or B=0, then fis
a Euclidean similarity and the image of the unit circle is a circle with center 0 and
radius |B|or |A|, respectively. Now we assume Aand Bare both nonzero. From
linear algebra we know that the image f(T)of the unit circle is an ellipse. We show
the ellipse f(T)has foci ±2√AB, semi-major axis with length |A|+|B|,andsemi-
minor axis with length |A|−|B|. A parametrization of the unit circle is t→ eit,so
the image of the unit circle is f(eit)=Aeit +Be−it =|A|ei(θ +t)+|B|ei(ϕ −t),where
A=|A|eiθand B=|B|eiϕ. The elementary inequalities
|A|−|B|≤|A|ei(θ+t)+|B|ei(ϕ−t)≤|A|+|B|,
show that the image ellipse contains the circle |z|=|A|−|B|and lies within the
circle |z|=|A|+|B|. Equality holds in the upper bound if and only if ei(θ +t)=ei(ϕ−t).
This occurs when θ+t=ϕ−t+2nπ,ort=1
2(ϕ −θ) +nπ,forsomen∈Z. Thus,
for t=1
2(ϕ −θ) and t=1
2(ϕ −θ) +π,|f(eit)|=|A|+|B|, so the length of the
semi-major axis is a=|A|+|B|.Sinceeiϕ/2=√B/|B|1/2and eiθ/2=√A/|A|1/2,
we obtain
fei(ϕ−θ)/2=|A|+|B|
|AB|1/2√AB .
Therefore, a direction vector for the major axis is √AB. Likewise, equality holds in
the lower bound when ei(θ+t)=−ei(ϕ −t);thatis,θ+t=ϕ−t+π+2nπ,ort=
1
2(ϕ −θ) +π
2+nπ,forsomen∈Z. Hence, the length of the semi-minor axis is
b=|A|−|B|.Thenc, the distance from the center of the ellipse to the foci, is
determined from c2=a2−b2=4|A||B|,sothatc=2|A|1/2|B|1/2. Consequently the
foci are ±2√AB.
October 2008] TRIANGLES, ELLIPSES, AND CUBIC POLYNOMIALS 683
4. THE STEINER INELLIPSE. For an equilateral triangle the inellipse is simply
the inscribed circle. The general case follows from this special case by making use of
affine transformations.
Proof of Steiner’s theorem. We begin by establishing the existence of an ellipse in-
scribed in a triangle and tangent at the midpoints of the sides. Let be the equilat-
eral triangle with vertices at 1, ω=exp(2πi/3)and ω2. Given any triangle z1z2z3
in C, there is a unique affine transformation f(z)=Az +B¯z+Cwith f(1)=z1,
f(ω) =z2and f(ω2)=z3. Straightforward calculations give
A=1
3(z1+ω2z2+ωz3),
B=1
3(z1+ωz2+ω2z3),
C=1
3(z1+z2+z3)=g.
Because the points z1,z2,z3are noncollinear, fis a bijection of C. The incircle for
is 1
2Tand is tangent at the midpoints of the sides, so f(1
2T)is an ellipse tangent to the
sides of z1z2z3at the midpoints since affine transformations preserve midpoints and
tangency. Note that
AB =1
9z2
1+z2
2+z2
3+(ω +ω2)(z1z2+z2z3+z1z3)
=1
9z2
1+z2
2+z2
3−(z1z2+z2z3+z1z3)
=1
9(z1+z2+z3)2−3(z1z2+z2z3+z1z3)
=g2−1
3(z1z2+z2z3+z1z3),
since 1 +ω+ω2=0. By Theorem 3.1, fmaps 1
2Tonto an ellipse with foci given by
(2.7).
Next, we establish uniqueness. Suppose Eis an ellipse inscribed in z1z2z3and
tangent at the midpoints of the sides. There is a bijective affine transformation gthat
maps the circle 1
2Tonto the ellipse E. Then the preimage of z1z2z3is a triangle, say
Z1Z2Z3, with incircle 1
2Tthat is tangent to the sides at the midpoints. Because both
tangent segments to a circle from an exterior point have the same length and the points
of tangency are the midpoints of the sides, Z1Z2Z3is equilateral. The circumcircle of
an equilateral triangle has double the radius of the incircle, so the vertices of Z1Z2Z3
lie on T. If necessary, we may precompose gwith a rotation to insure that Z1=1.
Then {Z2,Z3}={ω, ω2}; if necessary, we also precompose gwith a reflection over
the real axis to guarantee that Z2=ωand Z3=ω2.Theng=fand Eis the ellipse
constructed in the first part of the argument.
In passing we note that the ellipse f(T)passes through the vertices of z1z2z3and
has center at the centroid of the triangle; it is the unique ellipse with these properties
and is called the Steiner circumellipse.
An appealing byproduct of the proof of Steiner’s theorem is a known formula for
the area of a triangle in terms of the complex numbers representing its vertices. From
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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 115
the formulas for Aand Bwe obtain
|A|2−|B|2=2
3√3Im (z1¯z2+z2¯z3+z3¯z1).
The geometric interpretation of the determinant as a ratio of areas gives
|A|2−|B|2=|det(f)|=area z1z2z3
area ,
where det(f)denotes the determinant of the linear part Az +B¯zof f. From area =
3√3/4, we obtain the known formula
area z1z2z3=1
2Im (z1¯z2+z2¯z3+z3¯z1).
The incircle of a triangle has the largest radius, and so the largest area, among all
circles contained in the triangle. The analog of this property holds for the inellipse.
Theorem 4.1. Let C be any circle contained in z1z2z3.Then
area C
area z1z2z3≤π
3√3(4.1)
and equality holds if and only if z1z2z3is equilateral and C is the incircle.
Proof. Let rdenote the inradius of z1z2z3.ThenareaC≤πr2and equality holds if
and only if Cis the incircle. Therefore, it suffices to prove that
πr2
area z1z2z3≤π
3√3
with equality if and only if z1z2z3is equilateral. Let a,b,cdenote the lengths of
the sides of z1z2z3and s=1
2(a+b+c)the semiperimeter. By Heron’s formula
√s(s−a)(s−b)(s−c)is the area of z1z2z3; the product rs also gives the area.
Therefore,
r2s=area2z1z2z3
s=(s−a)(s−b)(s−c).
The arithmetic mean–geometric mean inequality gives
(s−a)(s−b)(s−c)1/3≤(s−a)+(s−b)+(s−c)
3=s
3
with equality if and only if a=b=c. Hence, r2≤s2/27, or r≤s/(3√3), and equal-
ity holds if and only if z1z2z3is equilateral. Then
πr2
area z1z2z3=πr
s≤π
3√3
with equality if and only if z1z2z3is equilateral.
October 2008] TRIANGLES, ELLIPSES, AND CUBIC POLYNOMIALS 685
Corollary 4.2 (Areal maximality of the inellipse). Among all ellipses contained in
a triangle the inellipse has the largest area. Precisely, for any ellipse E contained in
z1z2z3,
area E
area z1z2z3≤π
3√3(4.2)
and equality holds if and only if E is the inellipse.
Proof. First, let fbe the affine transformation with f(1)=z1,f(ω) =z2and
f(ω2)=z3.Then f(1
2T)=E0is the inellipse. Since an affine transformation scales
all areas by the same multiplicative factor,
area E0
area z1z2z3=area 1
2T
area =π
3√3.(4.3)
Now, consider any ellipse Econtained in z1z2z3. There is an affine transformation g
that maps the circle 1
2Tonto the ellipse E. The preimage of z1z2z3is a triangle, say
Z1Z2Z3, that contains 1
2T. Then by Theorem 4.1
area E
area z1z2z3=area 1
2T
area Z1Z2Z3≤π
3√3(4.4)
with equality if and only if Z1Z2Z3is equilateral and 1
2Tis the incircle. As in the
proof of Steiner’s theorem, the conditions for equality imply Eis the inellipse.
5. THE PROOF OF THEOREM 2.3. A proof of Theorem 2.3 is given in [2]; we
present a proof using complex numbers.
Proof of Theorem 2.3. Any line normal to the unit vector eiθhas an equation of the
form
xcos θ+ysin θ=Re (e−iθz)=c
for some c∈R.Sincedist(zj,) =Re e−iθzj−c,
D=
n
j=1Re e−iθzj−c2.(5.1)
The goal is to determine cand θso that Dis minimized. From
0=∂D
∂c=−
n
j=1
2Re e−iθzj−c,
we obtain
c=Re e−iθ1
n
n
j=1
zj.(5.2)
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THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 115
Thus, if a line produces an extreme value for D, then the centroid glies on .Ifwe
let wj=zj−g, then from (5.1) and (5.2) we obtain
D=
n
j=1
Re2e−iθwj.
By using the elementary identities Re(−iz)=Im zand Im(z2)=2(Re z)(Im z),we
find that
0=∂D
∂θ =2
n
j=1
Ree−iθwjRe−ie−iθwj
=2
n
j=1
Ree−iθwjIme−iθwj
=
n
j=1
Ime−2iθw2
j
=Im ⎛
⎝e−iθ
n
j=1
w2
j⎞
⎠
2
.
Thus, at a critical point of D,e−iθn
j=1w2
j2
is a real number, say t,andso
e−iθ
n
j=1
w2
j=√tif t≥0,
i√|t|if t<0. (5.3)
If t=0, or equivalently, n
j=1w2
j=0, then the function Dis constant and every line
through gis a line of best fit. The Second Derivative Test shows that t>0 corresponds
to a maximum value and t<0 to a minimum value. For t<0
ieiθ=
n
j=1w2
j
n
j=1w2
j
.(5.4)
Thus, when n
j=1w2
j= 0, the minimum value of Doccurs when is a line through g
that is parallel to the vector from 0 to n
j=1w2
j.
Example 5.1. If zj,1≤j≤3, are the vertices of an equilateral triangle, then every
line through the centroid is a line of best fit because Dis constant for lines through
the centroid. We determine the constant for the equilateral triangle with vertices
z1=1, z2=ω=exp(2πi/3)and z3=ω2.Ifis a line through the origin that is
perpendicular to eiθ,then
October 2008] TRIANGLES, ELLIPSES, AND CUBIC POLYNOMIALS 687
3
j=1
dist2(zj,) =
3
j=1
Re2e−iθe2πij/n
=3
2+Re e−2iθ
3
j=1
e4πij/n
=3
2
because Re2(z)=1
2|z|2+Re (z2). Thus, D=3
2.Itwouldbeusefultohavebetter
geometric insight as to why every line through the centroid of an equilateral triangle is
a line of best fit.
More generally, for the vertices of a regular polygon, any line through the centroid
is a line of best fit.
Example 5.2. It is not difficult to determine the line of best fit for the vertices of
an isosceles triangle; we state the result and leave the details to the reader. Let θ∈
(0,π)denote the vertex angle of an isosceles triangle. For narrow isosceles triangles
(θ∈(0,π/3)) the line of best fit is the median joining the vertex to the midpoint of
the base, while for wide isosceles triangles (θ∈(π/3,π)) it is the line through the
centroid that is parallel to the base. When θ=π/3 the isosceles triangle is equilateral
and every line through the center is a line of best fit.
ACKNOWLEDGEMENTS. The authors thank the referee for helpful comments and corrections. The au-
thors were partially supported by a subaward to the University of Cincinnati from the Institute for Advanced
Study through National Science Foundation grant EHR-0314808. The subaward supported a Math Science
Partnership with Cincinnati Public Schools.
Added in proof: An interesting, closely related article that employs a different method
recently appeared in the MONTHLY [3].
REFERENCES
1. L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979.
2. J. L. Coolidge, Two geometrical applications of the method of least squares, this MONTHLY 20 (1913)
187–190.
3. D. Kalman, An elementary proof of Marden’s theorem, this MONTHLY 115 (2008) 330–338.
4. M. Marden,
Geometry of Polynomials
, Mathematical Surveys No. 3, American Mathematical Society,
Providence, RI, 1966.
5. B. P. Palka, An Introduction to Complex Function Theory, Springer-Verlag, New York, 1991.
6. J. Siebeck, ¨
Uber eine neue analytische Behandlungweise der Brennpunkte, J. Reine Angew. Math. 64
(1864) 175–182.
7. J. Steiner,
Gesammelte Werke
, vol. 2, Prussian Academy of Sciences, Berlin, 1881–1882.
DAVI D MINDA is Charles Phelps Taft Professor of Mathematics at the University of Cincinnati. He grew up
in Cincinnati and received his B.S. (1965) and M.S. (1966) in mathematics from the University of Cincinnati.
He obtained his Ph.D. in 1970 from the University of California San Diego. After a year at the University of
Minnesota, he returned to the University of Cincinnati in 1971. His research interests are in geometric function
theory, especially the hyperbolic metric. He has been actively involved with both pre-service and inservice
math teachers. In 2001 he received the Dolly Cohen Award for Excellence in Teaching from the University
of Cincinnati and in 2002 the Ohio Section MAA Award for Distinguished College or University Teaching of
Mathematics.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221-0025
David.Minda@math.uc.edu
688 c
THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 115
STEVE PHELPS has taught mathematics at Madeira High School for sixteen years, where he is currently
the Chief Geometer and Head Football Coach. Born and raised in Cincinnati, he earned his B.S. in Secondary
Education (1991) and his M.A.T. in Teaching Mathematics (2001) from the University of Cincinnati. He is
haphazardly pursuing his Ed.D. His interests are in educational technologies, specifically, dynamic geometry
software and CAS (computer algebra systems). He is actively involved in the professional development of
mathematics teachers and the Park City Mathematics Institute. He is a regular presenter at conferences for
NCTM and Teachers Teaching with Technology.
Mathematics Department, Madeira High School, 7465 Loannes Drive, Cincinnati, OH 45243
sphelps@madeiracityschools.org
An Amendatory Limerick
Following the pattern of two previous communications to this MONTHLY [3, 1],
I submit the following claim:
−1=eiπ
Proves that Euler was doubtless a sly guy.
But ζ(2)
Was totally new
And raised the respect for him sky-high.
This claim is confirmed by a letter [2, p. 194] that James Stirling wrote to
Euler on April 13, 1738. It says:
“But most pleasing of all for me was your method for summing certain
series by means of powers of the circumference of the circle. I acknowledge
this to be quite ingenious and entirely new and I do not see that it has
anything in common with the accepted methods, so that I readily believe
that you have drawn it from a new source.”
REFERENCES
1. I. Grattan-Guiness, A limerick retort, this M ONTHLY 112 (2005) 232.
2. J. Tweddle, James Stirling, Scottish Academic Press, Edinburgh, 1988.
3. W. C. Willig, A limerick, this MONTHLY 111 (2004) 31.
—Submitted by William C. Waterhouse, Department of Mathematics,
Penn State University, University Park, PA 16802
October 2008] TRIANGLES, ELLIPSES, AND CUBIC POLYNOMIALS 689
- CitationsCitations8
- ReferencesReferences5
- "It is easy to improve on Theorem 5 using the following. Theorem 6 ([11], 2.3.) Suppose z j , 1 ≤ j ≤ n, are complex numbers, z VA is the centroid, and "
- "The Bôcher-Grace Theorem has been discovered independently by many mathematicians. Recently, proofs have been given by Kalman [3], and Minda and Phelps [6]. Maxime Bôcher proved the theorem in 1892 and then John H. Grace proved it in 1902. "
[Show abstract] [Hide abstract] ABSTRACT: The B\^{o}cher-Grace Theorem can be stated as follows: Let $p$ be a third degree complex polynomial. Then there is a unique inscribed ellipse interpolating the midpoints of the triangle formed from the roots of $p$, and the foci of the ellipse are the critical points of $p$. Here, we prove the following generalization: Let $p$ be an $n^{th}$ degree complex polynomial and let its critical points take the form $$ \alpha+\beta \cos k\pi/n, \quad k=1,...,n-1, \quad\beta\ne0. $$ Then there is an inscribed ellipse interpolating the midpoints of the convex polygon formed by the roots of $p$, and the foci of this ellipse are the two most extreme critical points of $p$: $\alpha\pm\beta \cos \pi/n$.- [Show abstract] [Hide abstract] ABSTRACT: If E is any ellipse inscribed in a convex quadrilateral, D, then we prove that Area(E)/Area(D) is less than or equal to pi/4, and equality holds if and only if D is a parallelogram and E is tangent to the sides of D at the midpoints. This extends well known results for ellipses inscribed in triangles. 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.
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