Alternative formulation and extensions Euler's quadrilateral theorem
euler s theorem parallelogram
euler derived theorem above corollary different theorem, 1 hand appears less elegant requires introduction of additional point on other hand provides more structural inside.
for given convex quadrilateral
a
b
c
d
{\displaystyle abcd}
euler introduced additional point
e
{\displaystyle e}
such
a
b
e
d
{\displaystyle abed}
forms parallelogram , following equality holds:
|
a
b
|
2
+
|
b
c
|
2
+
|
c
d
|
2
+
|
a
d
|
2
=
|
a
c
|
2
+
|
b
d
|
2
+
|
c
e
|
2
{\displaystyle |ab|^{2}+|bc|^{2}+|cd|^{2}+|ad|^{2}=|ac|^{2}+|bd|^{2}+|ce|^{2}}
the distance
|
c
e
|
{\displaystyle |ce|}
between additional point
e
{\displaystyle e}
, point
c
{\displaystyle c}
of quadrilateral not being part of parallelogram can thought of measuring how quadrilateral deviates parallelogram ,
|
c
e
|
2
{\displaystyle |ce|^{2}}
correction term needs added original equation of parallelogram law.
m
{\displaystyle m}
being midpoint of
a
c
{\displaystyle ac}
yields
|
a
c
|
|
a
m
|
=
2
{\displaystyle {\tfrac {|ac|}{|am|}}=2}
. since
n
{\displaystyle n}
midpoint of
b
d
{\displaystyle bd}
midpoint of
a
e
{\displaystyle ae}
,
a
e
{\displaystyle ae}
,
b
d
{\displaystyle bd}
both diagonals of parallelogram
a
b
e
d
{\displaystyle abed}
. yields
|
a
e
|
|
a
m
|
=
2
{\displaystyle {\tfrac {|ae|}{|am|}}=2}
, hence
|
a
c
|
|
a
m
|
=
|
a
e
|
|
a
m
|
{\displaystyle {\tfrac {|ac|}{|am|}}={\tfrac {|ae|}{|am|}}}
. therefore, follows intercept theorem (and reverse)
c
e
{\displaystyle ce}
,
n
m
{\displaystyle nm}
parallel ,
|
c
e
|
2
=
(
2
|
n
m
|
)
2
=
4
|
n
m
|
2
{\displaystyle |ce|^{2}=(2|nm|)^{2}=4|nm|^{2}}
, yields euler s theorem.
euler s theorem can extended larger set of quadrilaterals, includes crossed , nonplaner ones. holds called generalized quadrilaterals, consist of 4 arbitrary points in
r
n
{\displaystyle \mathbb {r} ^{n}}
connected edges form cycle graph.
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