1549055259-Ubiquitous_Quasidisk__The__Gehring_

3.12. QUADRILATERALS AND HARMONIC QUASISYMMETRY

Now suppose that Z1' Z2, Z3 is a triple of points in aB with

lz1 - z2I = lz2 - z3I

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and choose Z4 = -Z2. Then Z1' Z2' Z3' Z 4 is an ordered quadruple of points in aB

with [z1, z2, z3, z4] = 2, by elementary geometry.

We claim that

(3.12.6)

By symmetry it is sufficient to study the cases arg(z 2 ) E [O, 1f /2] and arg(z 2 ) E

[7r/2,1f]. In the first case arg(z 4 ) E [7r,31f/2] and arg(¢(z 4 )) E [1f,31f/2]. Moreover,

arg(¢(z2)) E [-1f/2,1f/2]. Since 1¢(1) +ii~ 2/a and¢ is order-preserving, we have

that l¢(z4) +ii ~ 2/a. This implies that l¢ (z2) - ¢ (z4)I ~ min{2/a, J2} = 2/ a.

The second case is proved in a similar way.

Next (3.12.6) implies that

2/a l¢(z1) - ¢(z3)I l¢(z1) - ¢(z3)l l¢(z2) - ¢(z4)I

• < <a
  2 l¢(z2) - ¢(z3)I - l¢ (z1) - ¢(z4)l l¢(z2) - ¢(z3)I - '
  whence
  l¢(z1) - ¢ (z2)I :S l¢(z1) - ¢(z3)I + l¢ (z2) - ¢ (z3)I :Sb l¢(z2) - ¢(z3)I

where b = a^2 + 1. This is the upper bound in (3.12.4).
Finally from Ptolemy's identity
(3.12.7)

we see that Z3,Z2,Z 1 ,Z4 is a quadruple of points in aB and with [z3,Z2,z1,z4] = 2.
Hence we can intercha nge the points z 1 and z 3 in the above argument to obtain the
lower bound in (3.12.4). D

We consider next a converse of Lemma 3.12.2. Our proof depends on the follow-
ing observation co ncerning the geometry of quadrilaterals Q in B with mod( Q) = 1.

REMARK 3.12.8. If Z1, z2, Z3, Z4 is an ordered quadruple of points in aB with

lz1 - z2llz3 - z4I = lz2 - z3llz4 - z1I,

lz1 - z2I :S min(lz2 - z3I, lz4 - z11),

then

lz3 - z4I < (J2 + 1) max(lz2 - z3I, lz4 - z11),

lz1 - z2I > (v'2 - 1) min(lz2 - z3I, lz4 - z11).

Both bounds are best possible.

PROOF. Let

a= lz1 - z2I,

Then by hypothesis,

ac = bd, a :::; min(b, d) :::; max(b, d) :::; c.

By interchanging z 1 , z 2 and z 3 , z 4 if necessary, we may also assume that

a:::; b:::; d:::; c.