Geometry • Topic 29

Spiral Similarity

A spiral similarity is a combination of a rotation and a homothety (scaling) centered at the same point. It is the natural transformation mapping one segment to another in the plane.

Definition

Definition (Spiral Similarity)

A transformation centered at

O

that maps

A \to A^{'}

and

B \to B^{'}

is a spiral similarity. It consists of:

Rotation: By angle $∠ A O A^{'}$ .
Homothety: With ratio $k = O A^{'} / O A$ .

Miquel Point

If we have four lines intersecting in general position, the circumcircles of the four triangles formed meet at a single point (the Miquel Point). This point is the center of a spiral similarity mapping specific segments.

Fundamental Theorem

A BC D

is a cyclic quadrilateral and lines

A B, C D

meet at

E

, then the center of the spiral similarity mapping

A C \to B D

is... (intersection of diagonals?). Actually: The center of spiral similarity mapping

A B \to C D

is the intersection of the circumcircles of

A D E

and

BCE

Practice Problems

Exercise (Problem 1)

Two squares

A BC D

and

A EFG

share a vertex

A

. Prove that

BE

and

D G

are perpendicular and equal in length. (Hint: Rotation by $9 0^{\circ}$ at $A$ is a spiral similarity).

Exercise (Problem 2)

Let two circles intersect at

A

and

B

. A line through

A

intersects the circles at

C

and

D

. Prove that the ratio

BC / B D

is constant.

Exercise (Problem 3)

Identify the center of the spiral similarity that maps a triangle to its medial triangle.