49. Newton’s Ellipse Problem
To determine the locus of the centers of all ellipses that can be inscribed in a
given (convex) quadrilateral.
Newton’s very elegant solution to this problem is based on the following theorem, also
from Newton:
Theorem. The line joining the centers of the diagonals of a quadrilateral
circumscribed about a circle passes through the center of the circle.
This follows from the
Lemma. The locus of the common vertex of two triangles with given bases and a
given area sum is a straight line.
Proof of the Lemma. Let p AB and q DE be the given bases, and x and y (length of)
altitudes from the common vertex C to the bases:
C
x
E
y
A
p
B
q
D
If K is the constant area sum, then
R
straight line.
1
2
px 1
2
qy K or px qy 2K, an equation of a
Proof of the Theorem. Let ABCD be a quadrilateral circumscribed about a circle with
center O and radius r.
1
A
d
a
N
B
D
O
M
b
c
C
Let the sides have length AB a, BC b, etc. as shown above. Since tangents to a
circle have the same length, it follows that a c b d. Let M and N be midpoints
of the diagonals AC and BD respectively, and let 2J be the area of the quadrilateral.
Let AŸd stand for the area of triangle d. Then
AŸdMAB AŸdMCD 1
2
1
2
ŸAŸdCAB AŸdACD Ÿ2J J,
and similarly for N. By the lemma, line MN is the locus of the common vertex of
pairs of triangles with bases AB and CD having a fixed area sum J. We claim that
AŸdOAB AŸdOCD J too. This puts O on line MN. We have
I
II
AŸdOAB AŸdOCD 12 ra 12 rc r a2c ,
AŸdOBC AŸdODA 12 rb 12 rd r b2d ,
and since a c b d, I II. But I II 2J, so I II J.
R
The solution to Newton’s problem. Consider any ellipse inscribed in the given
quadrilateral. This ellipse is the normal (perpendicular) projection of a circle. In this
projection, the quadrilateral is the image of a quadrilateral circumscribed about the
circle. Now
1.
2.
3.
the center of the circle lies on the line joining the midpoints of the
diagonals (by the theorem above),
normal projection ordinarily changes distance, but does preserve
midpoints, i.e., halving, and
the center of the ellipse is the image of the center of the circle.
Thus the locus of the centers of all ellipses that can be inscribed in a given
quadrilateral is the line segment joining the midpoints of the diagonals of the
quadrilateral.
2