User:Freddiezhao1/sandbox

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]

A spiral similarity taking triangle ABC to triangle A'B'C'.


Definition

edit

A spiral similarity   is composed of a rotation of the plane followed a dilation about a center   with coordinates   in the plane.[3] Expressing the rotation by a linear transformation   and the dilation as multiplying by a scale factor  , we find that a point   gets mapped to  


Relationship to Complex Numbers

edit

On the complex plane, any spiral similarity can be expressed in the form  , where   is a complex number. The magnitude   is the dilation factor of the spiral similarity, and the argument   is the angle of rotation. [4]

Properties

edit

Center of a Spiral Similarity for Two Line Segments

edit

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]

  • Draw lines   and  , and let   be the intersection of the two lines.
  • Draw the circumcircles of triangles   and  .
  • The circumcircles intersect at a second point  . Then   is the spiral center mapping   to  

 

Proof: Note that   and   are cyclic quadrilaterals. Thus,  . Similarly,  . Therefore, by AA similarity, triangles   and   are similar. Thus,   so a rotation angle mapping   to   also maps   to  . The dilation factor is then just the ratio of side lengths   to  .[3]

Solution with Complex Numbers

edit

If we express   and   as points on the complex plane with corresponding complex numbers   and  , we can solve for the expression of the spiral similarity which takes   to   and   to  . Note that   and  , so  . Since   and  , we plug in to obtain  , from which we obtain  .[3]

Pairs of Spiral Similarities

edit

For any points   and  , the center of the spiral similarity taking   to   is also the center of a spiral similarity taking   to  .

This can be seen through the above construction. If we let   be the center of spiral similarity taking   to  , then  . Therefore,  . Also,   implies that  . So by AA similarity we see that  , thus   is also the center of the spiral similarity which takes   to  .[3][4]

Corollaries

edit

Proof of Miquel's Quadrilateral Theorem

edit

Spiral similarity can be used to prove Miquel's Quadrilateral Theorem: given four noncollinear points   and  , the circumcircles of the four triangles   and   intersect at one point, where   is the intersection of   and   and   is the intersection of   and   (see diagram). [1]

 

Let   be the center of the spiral similarity which takes   to  . By the above construction, the circumcircles of   and   intersect at   and  . Since   is also the center of the spiral similarity taking   to  , by similar reasoning the circumcircles of   and   meet at   and  . Thus, all four circles intersect at  .[1]

Example Problem

edit

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

Given a scalene triangle  , let   and   be points on segments   and  , respectively, so that  . Let   be the circumcircle of triangle   and   the reflection of   across  . Lines   and   meet   again at   and  , respectively. Prove that   and   intersect on  .[3]

 

Proof: We first prove the following claims:

Claim 1: Quadrilateral   is cyclic.

Proof: Since   is isosceles, we note that   thus proving that quadrilateral   is cyclic, as desired. By symmetry, we can prove that quadrilateral   is cyclic.

Claim 2:  

Proof: We have that   By similar reasoning,   so by AA similarity,   as desired.

We now note that   is the spiral center that maps   to  . Let   be the intersection of   and  . By the spiral similarity construction above, the spiral center must be the intersection of the circumcircles of   and  . However, this point is  , so thus points   must be concyclic. Hence,   must lie on  , as desired.


  1. ^ a b c d Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. United States: MAA Press. pp. 196–200. ISBN 978-0-88385-839-4.
  2. ^ Coxeter, H.S.M. (1967). Geometry Revisited. Toronto and New York: Mathematical Association of America. pp. 95–100. ISBN 978-0-88385-619-2.
  3. ^ a b c d e Baca, Jafet (2019). "On a special center of spiral similarity". Mathematical Reflections. 1: 1–9.
  4. ^ a b Zhao, Y. (2010). Three Lemmas in Geometry.