Spiral similarity

Basic information

A spiral similarity is a plane transformation composed of a rotation of the plane and a dilation of the plane having the common center. The order in which the composition is taken is not important.

Any two directly similar figures are related either by a translation or by a spiral similarity (directly similar figures are similar and have the same orientation).

The transformation is linear and transforms any given object into an object homothetic to given.

On the complex plane, any spiral similarity can be expressed in the form $T(x) = x_0+k (x-x_0),$ where $k$ is a complex number. The magnitude $|k|$ is the dilation factor of the spiral similarity, and the argument $\arg(k)$ is the angle of rotation.

The spiral similarity is uniquely defined by the images of two distinct points. It is easy to show using the complex plane.

Let $A' = T(A), B' = T(B),$ with corresponding complex numbers $a', a, b',$ and $b,$ so $a' = T(a) = x_0 + k (a - x_0), b' = T(b) = x_0+ k (b-x_0) \implies$ $k = \frac {T(b) - T(a)}{b-a} = \frac {b' - a' }{b - a},$ $x_0=\frac {ab' - ba' }{a-a'+b' -b}, a' - a \ne b' - b.$ For any points $A, B, A',$ and $B'$ the center of the spiral similarity taking $AB$ to $A'B'$ point $x_{0}$ is also the center of a spiral similarity taking $A'B$ to $AB'.$ This fact explain existance of Miquel point.

Case 1 Any line segment $AB$ can be mapped into any other $A'B'$ using the spiral similarity. Notation is shown on the diagram. $P = AB \cap A'B'.$

$\Omega$ is circle $AA'P, \omega$ is circle $BB'P, x_0 = \Omega \cap \omega, x_0 \neq P,$

$C$ is any point of $AB, \theta$ is circle $CPx_0, C' = \theta \cap A'B'$ is the image $C$ under spiral symilarity centered at $x_0.$ $\triangle AA'x_0 \sim \triangle BB'x_0 \sim \triangle CC'x_0.$

$|k| = \frac {A'B'}{AB} = \frac {A'x_0}{Ax_0} = \frac {B'x_0}{Bx_0} = \frac {C'x_0}{Cx_0}$ is the dilation factor,

$\arg(k) =\angle APA'=\angle Ax_0A' =\angle Bx_0B' =\angle Cx_0C'$ is the angle of rotation.

Case 2 Any line segment $AB$ can be mapped into any other $BB'$ using the spiral similarity. Notation is shown on the diagram. $B = AB \cap BB', \Omega$ is circle $ABB$ (so circle is tangent to $BB'), \omega$ is circle tangent to $AB, x_0 = \Omega \cap \omega, x_0 \neq B, C$ is any point of $AB, \theta$ is circle $CBx_0,$ $C' = \theta \cap BB'$ is the image $C$ under spiral symilarity centered at $x_0.$ $\triangle ABx_0 \sim \triangle BB'x_0 \sim \triangle CC'x_0.$ $|k| = \frac {BB'}{AB}$ is the dilation factor,

$\angle Ax_0B = \arg(k)$ is the angle of rotation.

Simple problems

Explicit spiral symilarity

Given two similar right triangles $ABC$ and $A'B'C, k = \frac {AC}{BC},$ $\angle ACB = 90^\circ, D = AA' \cap BB'.$ Find $\angle ADB$ and $\frac {AA'}{BB'}.$

Solution

The spiral symilarity centered at $C$ with coefficient $k$ and the angle of rotation $90^\circ$ maps point $B$ to point $A$ and point $B'$ to point $A'.$

Therefore this symilarity maps $BB'$ to $AA' \implies$ $\frac {AA'}{BB'} = k, \angle ADB = 90^\circ.$

vladimir.shelomovskii@gmail.com, vvsss

Hidden spiral symilarity

Let $\triangle ABC$ be an isosceles right triangle $(AC = BC).$ Let $S$ be a point on a circle with diameter $BC.$ The line $\ell$ is symmetrical to $SC$ with respect to $AB$ and intersects $BC$ at $D.$ Prove that $AS \perp DS.$

Proof

Denote $\angle SBC = \alpha, \angle SCB = \beta = 90^\circ - \alpha,$ $\angle SCA = \alpha, \angle BSC = 90^\circ, k = \frac {SC}{SB} = \cot \beta.$ Let $SC$ cross perpendicular to $BC$ in point $B$ at point $D'.$

Then $\frac {BC}{BD'} = \cot \beta.$

Points $D$ and $D'$ are simmetric with respect $AB,$ so $BD = BD' \implies k = \frac {SC}{SB} = \frac {BC}{BD}.$

The spiral symilarity centered at $S$ with coefficient $k$ and the angle of rotation $90^\circ$ maps $B$ to $C$ and $D$ to point $D_0$ such that $k \cdot BD_0 = BC = AC, \angle D_0CS = \angle DBS \implies D_0 = A.$

Therefore $\angle ASC = \angle DSB \implies$ $\angle ASD = \angle ASC - \angle DSC = \angle DSB - \angle DSC = \angle BSC = 90^\circ.$ vladimir.shelomovskii@gmail.com, vvsss

Linearity of the spiral symilarity

$\triangle ABF \sim \triangle BCD \sim \triangle CAE.$ Points $D,E,F$ are outside $\triangle ABC.$

Prove that the centroids of triangles $\triangle ABC$ and $\triangle DEF$ are coinsite.

Proof

Let $\vec y = T(\vec x),$ where $T$ be the spiral similarity with the rotation angle $\angle BAF= \angle CBD = \angle ACE$ and $k = \frac {|AF|}{|AB|} = \frac {|DB|}{|BC|} = \frac {|EC|}{|CA|}.$

A vector has two parameters, modulo and direction. It is not tied to a center of the spiral similarity. Therefore $\vec AF = T(\vec AB), \vec BD = T(\vec BC), \vec CE = T(\vec CA).$ $\vec AB + \vec BC + \vec CA = \vec 0.$ We use the property of linearity and get $\vec AF + \vec BD + \vec CE = k(\vec AB + \vec BC + \vec CA) = k \cdot \vec 0 = \vec 0.$ Let $G$ be the centroid of $\triangle ABC$ so $\vec GA + \vec GB + \vec GC = \vec 0 \implies$

$\vec GD + \vec GE + \vec GF = \vec 0 \implies G$ is the centroid of the $\triangle DEF.$

vladimir.shelomovskii@gmail.com, vvsss

Construction of a similar triangle

Let triangle $\triangle ABC$ and point $A'$ on sideline $BC$ be given. Construct $\triangle A'B'C' \sim \triangle ABC$ where $B'$ lies on sideline $AC$ and $C'$ lies on sideline $AB.$

Solution

Let $T(X)$ be the spiral symilarity centered at $A'$ with the dilation factor $k = \frac {AB}{AC}$ and rotation angle $\alpha = \angle BAC, A_1 = T(A), B_1 = T(B).$

$A_1B_1 = T(AB)$ so image of any point $C' \in AB$ lies on $A_1B_1.$ $B' \in AC \implies B' = A_1B_1 \cap AC.$ The spiral symilarity $T^{-1}(X)$ centered at $A'$ with the dilation factor $k^{-1} = \frac {AC}{AB}$ and rotation angle $-\alpha$ maps $B'$ into $C'$ and $\angle B'A'C' = \alpha, \frac {A'C'}{A'B'} = k^{-1} = \frac {AC}{AB}$ therefore the found triangle $\triangle A'B'C' \sim \triangle ABC$ is the desired one.

vladimir.shelomovskii@gmail.com, vvsss

Center of the spiral symilarity for similar triangles

Let triangle $\triangle ABC$ and point $C' (C' \ne C, C' \ne B)$ on sideline $BC$ be given. $\triangle A'B'C' \sim \triangle ABC$ where $B'$ lies on sideline $AB$ and $A'$ lies on sideline $AC.$ The spiral symilarity $T$ maps $\triangle ABC$ into $\triangle A'B'C'.$ Prove

a) $\angle AB'A' = \angle BC'B' = \angle CA'C'.$

b) Center of $T$ is the First Brocard point of triangles $\triangle ABC$ and $\triangle A'B'C'.$

Proof

a) Let $\tau(X)$ be the spiral symilarity centered at $C'$ with the dilation factor $k = \frac {AC}{BC}$ and rotation angle $\gamma = \angle ACB, A_1 = \tau(A), B_1 = \tau(B).$ $A' = A_1B_1 \cap AC, B' = \tau^{-1}(A').$

Denote $\varphi = \angle BC'B' \implies \angle CC'A' = 180^\circ - \varphi - \gamma \implies$ $\angle CA'C' = 180^\circ - \gamma - \angle CC'A' = \varphi.$ Similarly $\angle AB'A' = \varphi.$

b) It is well known that the three circumcircles $AA'B', BB'C',$ and $CC'A'$ have the common point (it is $D$ in the diagram).

Therefore $AB'DA'$ is cyclic and $\angle AB'A' = \angle ADA' =\varphi.$

Similarly, $\angle BDB' = \angle CDC' = \varphi.$

$\angle A'DC' = 180^\circ - \gamma, \angle A'DC = 180^\circ - \gamma - \varphi \implies$ $\angle ADC = \angle A'DC' = 180^\circ - \gamma.$ $\angle CAD + \angle ACD = 180^\circ - \angle ADC = \gamma = \angle ACD + \angle BCD \implies \angle CAD = \angle BCD = \psi.$

Similarly, $\angle CAD = \angle ABD = \angle BCD = \psi.$

Therefore, $D$ is the First Brocard point of $\triangle ABC.$

$AB'DA'$ is cyclic $\implies \angle A'B'D = \angle A'AD = \psi.$ Similarly, $\angle B'C'D = \angle C'A'D = \psi.$

Therefore $D$ is the First Brocard point of $\triangle A'B'C',$ and $\triangle A'DC' \sim \triangle ADC, \triangle A'DB' \sim \triangle ADB.$

Therefore the spiral symilarity $T$ maps $\triangle ABC$ into $\triangle A'B'C'$ has the center $D,$ the angle of the rotation $\varphi.$

vladimir.shelomovskii@gmail.com, vvsss

Common point for 6 circles

Let $\triangle ABC$ and point $A'$ on sideline $BC$ be given. $\triangle A'B'C' \sim \triangle ABC$ where $B'$ lies on sideline $AC$ and $C'$ lies on sideline $AB.$

Denote $D = BB' \cap CC', E = AA' \cap CC', F = BB' \cap AA'.$

Prove that circumcircles of triangles $\triangle ABF, \triangle A'B'F, \triangle BCD,$ $\triangle B'C'D, \triangle ACE, \triangle A'C'E$ have the common point.

Proof

$\triangle A'B'C' \sim \triangle ABC$ so there is the spiral symilarity $T$ taking $\triangle ABC$ to $\triangle A'B'C', T(ABC) = A'B'C'.$ Denote $O$ the center of $T.$ $A'B' = T(AB), A'B = \tau(AB'), F = AA' \cap BB' \implies$ the center of $\tau$ is the secont crosspoint of circumcircles of $\triangle ABF$ and $\triangle A'B'F,$ but this center is point $O,$ so these circles contain point $O$ . Similarly for another circles.

vladimir.shelomovskii@gmail.com, vvsss

Three spiral similarities

Let triangle $\triangle ABC$ be given. The triangle $\triangle ACE$ is constructed using a spiral similarity of $\triangle ABC$ with center $A$ , angle of rotation $\angle BAC$ and coefficient $\frac {AC}{AB}.$

A point $D$ is centrally symmetrical to a point $B$ with respect to $C.$

Prove that the spiral similarity with center $E$ , angle of rotation $\angle ACB$ and coefficient $\frac {BC}{AC}$ taking $\triangle ACE$ to $\triangle CDE.$

Proof

$\angle ECD = 180^\circ - \angle ACB - \angle ABC = \angle BAC.$ $EC = BC \cdot \frac {AC}{AB}, CD = BC = AC \cdot \frac {BC}{AC} \implies \frac {CD}{EC}=\frac {AB}{AC} \implies \triangle CDE \sim \triangle ACE \implies \angle DEC = \angle CEA.$

Corollary

Three spiral similarities centered on the images of the vertices of the given triangle $\triangle ABC$ and with rotation angles equal to the angles of $\triangle ABC$ take $\triangle ABC$ to $\triangle FDC$ centrally symmetric to $\triangle ABC$ with respect to $C.$

vladimir.shelomovskii@gmail.com, vvsss

Superposition of two spiral similarities

Let $T$ be the spiral similarity centered at $A$ with angle $\alpha$ and coefficient $\frac {1}{|k|}, Y = T(X), \frac {XA}{YA} = k.$

Let $t$ be spiral similarity centered at $B$ with angle $\alpha$ and coefficient $k, Z = t(Y), \frac {BZ}{BY} = k.$

Let $t(T(O)) = O.$

Prove: a) $XO = ZO, AO = BO, O$ is the crosspoint of bisectors $AB$ and $XZ.$

b) $\angle XOZ = 2 \alpha, \angle AOB = 2 \angle AXY.$

Algebraic proof

We use the complex plane $x = \vec X, y = \vec Y, z = \vec Z, a = \vec A,b = \vec B, o = \vec O.$ $y = T(x) = a + \frac {e^{i \alpha}}{k}(x - a), z = t(y) = b + k e^{i \alpha}(y - b) \implies$ $z = e^{2 i \alpha}x + b(1 - k e^{i \alpha}) + a e^{i \alpha}(k - e^{i \alpha}) = o + e^{2 i \alpha}(x - o).$ Let $a = (-1,0), b = (1,0).$ Then $o = i(\cot \alpha - \frac {k}{\sin \alpha}) \implies XO = ZO, AO = BO, \angle XOZ = 2 \alpha, \angle AOB = 2 \angle AXY.$ Geometric proof

Denote $\angle XAY = \alpha, \angle AXY = \beta, \angle AYX = \gamma = 180^\circ - \alpha - \beta.$

Then $k = \frac {XA}{YA} = \frac {\sin \gamma}{\sin \beta}, \triangle BZY \sim \triangle AXY.$

Let $H$ be the midpoint $AB, O$ be the point on bisector $AB$ such that $\angle AOH = \beta, C$ be the point on bisector $AB$ such that $\angle OAC = \alpha.$ Then

$\triangle AOC = \triangle BOC \sim \triangle AXY \sim \triangle BZY \implies$ $k = \frac {OA}{CA}, k^{-1} = \frac {CB}{OB},\angle CBO =\alpha \implies$ $T(O) = C, t(C) = O, t(T(O)) = O, \angle AOB = 2 \beta.$ $T(XO) = YC \implies \frac {XO}{YC} = k; t(YC) = ZO \implies \frac {ZO}{YC} = k \implies$ $XO = ZO, \angle (XO)(YC) = \angle (YC)(ZO) = \alpha \implies \angle (XO)(ZO) = \angle XOZ = 2 \alpha.$ $XO = ZO, AO = BO \implies O$ is the crosspoint of bisectors $AB$ and $XZ.$

Corollary

There is another pare of the spiral similarities centered at $X$ and $Z$ with angle $\beta,$ coefficients $k' = \frac {XA}{XY} = \frac {\sin \gamma}{\sin \alpha}$ and $k'^{-1}, Y = T'(A), B = t'(Y).$

In this case $t'(T'(O)) = O.$

vladimir.shelomovskii@gmail.com, vvsss

Spiral similarity for circles

1. Let circle $\omega$ cross circle $\Omega$ at points $A$ and $B.$ Point $C$ lies on $\omega.$

Spiral similarity $T$ centered at $A$ maps $\omega$ into $\Omega: \Omega = T(\omega), C' = T(C).$

Prove that points $C, C',$ and $B$ are collinear.

Proof

Arcs $\overset{\Large\frown} {AC} = \overset{\Large\frown} {ABC'}, \angle ABC = \frac {\overset{\Large\frown} {AC}}{2},$ $\angle ABC' = \frac {360^\circ -\overset{\Large\frown} {ABC'}}{2} = 180 ^\circ - \frac {\overset{\Large\frown} {ABC'}}{2} = 180 ^\circ - \angle ABC' \implies$ $B \in CC'.$

Corollary

Let $C \in \omega, C' \in \Omega,$ points $C, C',$ and $B$ be collinear.

Then exist the spiral similarity $T$ centered at $A$ such that $T(\omega) = \Omega, T(C) = C'.$

2. Let circle $\omega$ cross circle $\Omega$ at points $A$ and $B.$

Points $C$ and $D$ lie on $\omega, C' = BC \cap \Omega, D' = BD \cap \Omega.$

Let $EA$ be the tangent to $\omega, FA$ be the tangent to $\Omega.$

Prove that angle between tangents is equal angle between lines $CD$ and $C'D'.$

Proof

There is the spiral similarity $T$ centered at $A$ such that $T(\omega) = \Omega, T(C) = C', T(D) = D'.$ Therefore $T(CD) = C'D', T(EA) = FA \implies$ angles between these lines are the same.

vladimir.shelomovskii@gmail.com, vvsss

Remarkable point of spiral similarity

Circles $\omega$ and $\Omega$ centered at points $O$ and $O',$ respectively intersect at points $A$ and $B.$ Points $C \in \omega, C' \in \Omega,$ and $A$ are collinear. Point $X$ is symmetrical to $A$ with respect to the midpoint $OO'$ point $M.$ Prove:

a) $\angle ABX = 90^\circ,$

b) $CX = C'X.$

Proof

a) $AB \perp OO', OO'$ cross $AB$ in midpoint $\implies BX || OO' \implies \angle ABX = 90^\circ.$

b) $OM = O'M, AM = MX \implies AOXO'$ is parallelogramm $\implies AO = XO' = CO, OX = AO' = C'O'.$

Denote $\angle OAC = \angle OCA = \alpha, \angle O'AC' = \angle O'C'A = \beta, \angle XO'A = \angle XOA = \varphi \implies$

$180^\circ = \alpha + (180^\circ - \varphi) + \beta \implies \alpha + \beta = \varphi.$ $\angle COX = 360^\circ - (180^\circ - 2 \alpha) - \varphi = 180^\circ + \alpha - \beta.$ $\angle XO'C' = \varphi + \angle AO'C' = \varphi + (180^\circ - 2 \beta) = 180^\circ + \alpha - \beta = \angle COX \implies \triangle COX = \triangle XO'C' \implies CX = C'X.$

vladimir.shelomovskii@gmail.com, vvsss

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Spiral similarity

Contents

Basic information

Simple problems

Explicit spiral symilarity

Hidden spiral symilarity

Linearity of the spiral symilarity

Construction of a similar triangle

Center of the spiral symilarity for similar triangles

Common point for 6 circles

Three spiral similarities

Superposition of two spiral similarities

Spiral similarity for circles

Remarkable point of spiral similarity