Spiral similarity

Spiral similarity is a plane transformation in mathematics composed of a rotation and a dilation.

[1] It is used widely in Euclidean geometry to facilitate the proofs of many theorems and other results in geometry, especially in mathematical competitions and olympiads.

Though the origin of this idea is not known, it was documented in 1967 by Coxeter in his book Geometry Revisited.

[2] and 1969 - using the term "dilative rotation" - in his book Introduction to Geometry.

[3] The following theorem is important for the Euclidean plane: Any two directly similar figures are related either by a translation or by a spiral similarity.

is composed of a rotation of the plane followed a dilation about a center

[5] Expressing the rotation by a linear transformation

and the dilation as multiplying by a scale factor

On the complex plane, any spiral similarity can be expressed in the form

is the dilation factor of the spiral similarity, and the argument

[6] Let T be a spiral similarity mapping circle k to k' with k

Remark: This property is the basis for the construction of the center of a spiral similarity for two line segments.

, as rotation and dilation preserve angles.

Through a dilation of a line, rotation, and translation, any line segment can be mapped into any other through the series of plane transformations.

We can find the center of the spiral similarity through the following construction:[1]

so a rotation angle mapping

The dilation factor is then just the ratio of side lengths

, we can solve for the expression of the spiral similarity which takes

, the center of the spiral similarity taking

is also the center of a spiral similarity taking

be the center of spiral similarity taking

{\displaystyle {\frac {AX}{BX}}={\frac {CX}{DX}}}

{\displaystyle {\frac {AX}{CX}}={\frac {BX}{DX}}}

is also the center of the spiral similarity which takes

[5][6] Spiral similarity can be used to prove Miquel's quadrilateral theorem: given four noncollinear points

{\displaystyle \triangle PAB,\triangle PDC,\triangle QAD,}

be the center of the spiral similarity which takes

is also the center of the spiral similarity taking

Here is an example problem on the 2018 Japan MO Finals which can be solved using spiral similarity:Given a scalene triangle

By symmetry, we can prove that quadrilateral