A Family of Conics Associated with
the Configuration of Pappus
Leroy J. Dickey
 Figure 1. The conic Q(F).
 Figure 2.
The three conics Q(D), Q(E) and Q(F)
meet L_{1} at T_{1} and T_{2}.
 Figure 3.
All six conics: Q(A) Q(B), Q(C), Q(D), Q(E) and Q(F).
 Figure 4.
A Steiner conic associated
with the configuration of Pappus.
In this paper, we show a family of six conics
connected with the Configuration of Pappus.
These conics meet in interesting ways. There are two points
on one of the base lines that are common to three of the
conics. There are two points on the other base line
that are common to the other three conics. There are
two further points that are common to all six conics.
These six points together with the intersection of the
two base lines form a complete quadrangle.
In the second part of this paper, we show the connection
between the above six conics and 12 Steiner conics
that are connected with the configuration of Pappus.
 The Base Figure
 Let L_{1} and L_{2} be two distinct (base) lines.
 Let O be the point of intersection of the two base
lines L_{1} and L_{2}.
 Let A, B and C be three points on the line L_{1}
that are distinct from each other and distinct from O.
 Let D, E and F be three points on the line L_{2}
that are distinct from each other and distinct from O.
 The Six Conics
We construct six conics, Q(A), Q(B) Q(C), Q(D) Q(E) and Q(F) as follows:
 Let S = { A, B, C, D, E, F }.
 For each point X in S, construct a conic Q(X) as follows:
 Let S(X) be the five points in S \ { X }.
 Let S_{0}(X) be the set of three
points in S(X) that are on the line
(either L_{1} or L_{2})
not containing X.
 Let S_{0}(X) = S(X)  S_{1}(X).
 Construct the six lines connecting the 3 points in
S_{1}(X) to the 2 points in S_{2}(X).
 These six lines meet in eleven distinct points.
Five of the eleven are in S.
The remaining six points lie on a conic
that we call Q(X),
for X = A, B, C, D, E, or F.
 ( Figure 1 shows the particular conic Q(F). )
With reference to the Base Figure and the
Six Conics, the following properties hold:
 There are two points,
T_{1} and T_{2}
on the line
L_{1}
that the lie on all three conics
Q(D), Q(E) and Q(F).
 There are two points,
U_{1} and U_{2}
on the line
L_{2}
that lie on all three conics
Q(A), Q(B) and Q(C).
 There are two points,
V_{1} and V_{2},
that are diagonal points of the complete quadrangle
T_{1}, T_{2}, U_{1}, U_{2}
and lie on all six conics.
 Set up the coordinates for the Base Figure.
 Find the equations of the
Six Conics
Q(A), Q(B), Q(C), Q(D), Q(E) and Q(F).
 Find the coordinates of T_{1} and T_{2},
the two points of intersection of L_{1} with Q(F).
 Verify that points T_{1} and T_{2} are on the conic Q(D).
 Verify that points T_{1} and T_{2} are on the conic Q(E).
 ( Figure 2 shows three conics Q(D), Q(E) and Q(F) and how they
meet at the two points
T_{1} and T_{2} on L_{1}. )
Figure 2: The three conics Q(D), Q(E) and Q(F)
meet L_{1} at T_{1} and T_{2}.
More about Figure 2.
List of Figures
 Find the coordinates of U_{1} and U_{2},
the two points of intersection of L_{2} with the conic Q(C).
 Verify that points U_{1} and U_{2} are on the conic Q(A).
 Verify that points U_{1} and U_{2} are on the conic Q(B).
 Let V_{1} be the point of intersection of lines
T_{1} U_{1} and T_{2} U_{2}
 Let V_{2} be the point of intersection of lines
T_{1} U_{2} and T_{2} U_{1}
 Verify that both V_{1} and V_{2}
are on all Six Conics.
 ( Figure 3 shows the Six Conics
and how they meet by threes at
T_{1}, T_{2} and
U_{1}, U_{2},
and how they all meet at
V_{1} and V_{2}. )
The proof of the theorem is analytic and uses the symbolic algebra
programming language MAPLE V.
 The data (source)
This file contains the Maple V source code
giving the coordinates
for the six points of the base figure.
All other calculations are based on these six points.
 The functions (source)
This file contains the Maple V source code
for seventeen functions that are
used to construct all the other objects of this
construction, whether they be points, lines or conics.
 The defined objects
 The Maple V
source code
for the constructions.
 The
new objects
consist of the equations of conics and coordinates
of new lines and points.
 The proof (validation of incidences).
 The Maple V
source code
for the validations.
 The results.
The results of the validation section.
All claimed incidences are as they should be.
Given any six distinct
points points X, Y, Z, U, V, W for which X, Y and Z
are on one line and U, V and W are on the another,
and all six are distinct from the intersection point of the two lines,
the three points
YW.ZV,
ZU.XW and
XV.YU
lie on a line denoted by P(X,Y,Z; U,V,W).
 S_{e}, the "even" Steiner point.
 The three Pappus lines
 P(A,B,C; D,E,F),
 P(A,B,C; E,F,D) and
 P(A,B,C; F,D,E)
coincide at a point denoted by S_{e}.
 The subscript "e" is used because the ordered sequences
(D,E,F), (F,D,E) and (E,F,D) are the three even
permutations of the three symbols D, E and F.
 S_{o}, the "odd" Steiner point.
 The three Pappus lines
 P(A,B,C; F,E,D),
 P(A,B,C; D,F,E) and
 P(A,B,C; E,D,F)
coincide at a point denoted by S_{o}.
 The subscript "o" is used because the ordered sequences
(F,E,D), (D,F,E) and (E,D,F) are the three odd
permutations of the three symbols D, E and F.
Both of the points
S_{e} and S_{o}
depend on all six points in S = {A, B, C, D, E, F}.
For X in S, the locus of
S_{e}, as a function of X, is called
Q_{e}(X).
Similarly, the locus of
S_{o},
as a function of X, is called
Q_{o}(X).
We outline here the proof that
Q_{e}(X) and Q_{e}(X)
are conics.
For simplicity, we deal specifically with the particular choice X=F,
and the other cases for A, B, C, D and E are similar.

Consider F as a control point (parameter) allowed to
move along on the entire line L_{2}.

As a function of F, The locus of S_{o},
is a conic
that contains the
five distinct points D, E, AE.BD, BE.CD, CE.AD, no three of
which are collinear.
 This conic is denoted by Q_{o}(F) and
is defined by the five points D, E, AE.BD, BE.CD, CE.AD .
 Figure 4 shows this conic and three Pappus lines
meeting at the Steiner point S_{0}.
Figure 4. A Steiner conic S_{e}(F)
associated
with the configuration of Pappus.
More about Figure 4.
List of Figures

Similarly, the three Pappus lines
P(A,B,C; D,F,E),
P(A,B,C; F,E,D) and
P(A,B,C; E,D,F)
meet at a second Steiner point, S_{1},
whose locus, as a function of F, is a second conic
that contains the five points D, E, AD.BE, BD.CE, CD.AE.

This locus is the conic defined by the
five points D, E, AD.BE, BD.CE, CD.AE .

Take the union of the two sets of five points discovered above,
{ D, E, AE.BD, BE.CD, CE.AD } and
{ D, E, AD.BE, BD.CE, CD.AE }
and remove the two points of S_{2}(F), (namely D and E),
to obtain the set of six points
{ AE.BD, BE.CD, CE.AD,
AD.BE, BD.CE, CD.AE }.
All six of these points lie on the conic Q(F).
[Pascal]
.
There is a connection between the six conics Q(A), Q(B), Q(C),
Q(D), Q(E) and Q(F) (defined above),
 In the same way, for any X in the set S,
the loci of the two Steiner points S_{0} and S_{1},
as a function of X,
(the conics Q_{0}(X) and Q_{1}(X), respectively)
lead us to six interesting points,
any five of which determine / characterize the conic Q(X).
This brings us full circle back to the beginning
of the paper.
The end.
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Email to
ljdickey@math.uwaterloo.ca.
© 1997, 1998, 2003 Leroy J. Dickey