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Figure 1: D(ω) (solid lines) and D(ω ∪ {x}) \ D(ω) (dashed lines). The difference region ∆x(ω) isshaded in grey.
Figure 2: The set Cx(ω) for the configuration ω of Fig. 1, with a tile τ∗ ∈ C(0)x (ω) (light grey), itscircumcircle B(τ∗) (dashed), the associated edges ei and regions Wi (dark grey), and two trianglesτi ∈ Cx(ω) with τi ⊂ Wi with their circumcircles (solid).The construction in the proof gives˜I(τ∗) = τ2.

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