Sec. 10.9 Some results on circles 167
This is an identity due to Euler and from it
(z−z 4 )(z 2 −z 3 )
(z−z 3 )(z 2 −z 4 )+
(z−z 2 )(z 4 −z 3 )
(z−z 3 )(z 4 −z 2 )= 1. (10.9.2)
By 10.9.3 both fractions on the left are real-valued. AsZandZ 3 are on opposite
sides ofZ 2 Z 4 , there is a pointWof[Z,Z 3 ]onZ 2 Z 4 .ThenWis an interior point of the
circle, and soW∈[Z 2 ,Z 4 ]as the only points of the lineZ 2 Z 4 which are interior to the
circle are in this segment. It follows thatZ 2 andZ 4 are on opposite sides ofZZ 3 .Then
[Z,Z 2 ],[Z 3 ,Z 4 ]are in different closed half-planes with common edge the lineZZ 3 ,so
they have no points in common. It follows thatZandZ 2 are on the one side ofZ 3 Z 4
so the first of the fractions in (10.9.2) is positive, and so equal to its own absolute
value. But[Z,Z 4 ]and[Z 3 ,Z 2 ]are in different closed half-planes with common edge
ZZ 3 so they have no point in common. It follows thatZandZ 4 are on the one side of
Z 2 Z 3 , so the second fraction in (10.9.2) is positive and so equal to its own absolute
value. Hence
|(z−z 4 )(z 2 −z 3 )|
|(z−z 3 )(z 2 −z 4 )|
+
|(z−z 2 )(z 4 −z 3 )|
|(z−z 3 )(z 4 −z 2 )|= 1.
This is known asPtolemy’s theorem
From the original identity (10.9.2) withZ 1 replacingZwe can deduce that for four
distinct pointsZ 1 ,Z 2 ,Z 3 ,Z 4
Z 1 Z (^4) F
Z 1 Z 2 F
Z 2 Z (^3) F
Z 2 Z 4 F
+
Z 1 Z (^2) F
Z 1 Z 3 F
Z 4 Z (^3) F
Z 4 Z 2 F
= 1.
This can be expanded as
|Z 1 ,Z 4 |
|Z 1 ,Z 3 |cisα|Z 2 ,Z 3 |
|Z 2 ,Z 4 |
cisβ+|Z 1 ,Z 2 |
|Z 1 ,Z 3 |
cisγ|Z 4 ,Z 3 |
|Z 2 ,Z 4 |
cisδ= 1 ,where
α=FZ 3 Z 1 Z 4 ,β=FZ 4 Z 2 Z 3 ,γ=FZ 3 Z 1 Z 2 ,δ=FZ 4 Z 2 Z 3.From this we have that
|Z 1 ,Z 4 |
|Z 1 ,Z 3 ||Z 2 ,Z 3 |
|Z 2 ,Z 4 |
cis(αF+βF)+|Z 1 ,Z 2 |
|Z 1 ,Z 3 |
|Z 4 ,Z 3 |
|Z 2 ,Z 4 |
cis(γF+δF)= 1.We get two relationships on equating the real parts in this and also equating the
imaginary parts.
NOTE. For other applications of complex numbers to geometry, see Chapter 11
and Hahn [8].