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Difference between revisions of "Isogonal conjugate"

(→‎Definition of isogonal conjugate of a point)
(→‎The isogonal theorem)
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It is clear that, under the reverse transformation (also projective), such pairs of lines become isogonals, and the points equidistant from <math>\ell</math> lie on the isogonals.
 
It is clear that, under the reverse transformation (also projective), such pairs of lines become isogonals, and the points equidistant from <math>\ell</math> lie on the isogonals.
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<i><b>The isogonal theorem</b></i>
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Let two pairs of isogonals <math>OX – OX'</math> and <math>OY – OY'</math> be given. Let lines <math>XY</math> and <math>X'Y'</math> intersect at point <math>Z.</math> Let lines <math>X'Y</math> and <math>XY'</math> intersect at point <math>Z'.</math> Prove that <math>OZ</math> and <math>OZ'</math> are the isogonals with respect to the pair <math>(\ell,O).</math>
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<i><b>Proof</b></i>
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Let us perform a projective transformation of the plane that maps the point <math>O</math> into a point at infinity and the line <math>\ell</math> maps to itself. In this case, the isogonals turn into a pair of straight lines parallel to <math>\ell</math> and equidistant from <math>\ell.</math>
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The reverse (also projective) transformation maps the points equidistant from <math>\ell</math> onto isogonals.
  
 
'''vladimir.shelomovskii@gmail.com, vvsss'''
 
'''vladimir.shelomovskii@gmail.com, vvsss'''
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== Definition of isogonal conjugate of a point ==
 
== Definition of isogonal conjugate of a point ==
 
[[File:Definitin 1.png|400px|right]]
 
[[File:Definitin 1.png|400px|right]]

Revision as of 15:09, 26 February 2023

Isogonal conjugates are pairs of points in the plane with respect to a certain triangle.

The isogonal theorem

Isogonal lines definition

Let a line $\ell$ and a point $O$ lying on $\ell$ be given. A pair of lines symmetric with respect to $\ell$ and containing the point $O$ be called isogonals with respect to the pair $(\ell,O).$

Sometimes it is convenient to take one pair of isogonals as the base one, for example, $OA$ and $OB$ are the base pair. Then we call the remaining pairs as isogonals with respect to the angle $\angle AOB.$

Projective transformation

It is known that the transformation that maps a point with coordinates $(x,y)$ into a point with coordinates $(\frac{1}{x}, \frac {y}{x}),$ is projective.

If the abscissa axis coincides with the line $\ell$ and the origin coincides with the point $O,$ then the isogonals define the equations $y = \pm kx,$ and the lines $(\frac{1}{x}, \pm k)$ symmetrical with respect to the line $\ell$ become their images.

It is clear that, under the reverse transformation (also projective), such pairs of lines become isogonals, and the points equidistant from $\ell$ lie on the isogonals.

The isogonal theorem

Let two pairs of isogonals $OX – OX'$ and $OY – OY'$ be given. Let lines $XY$ and $X'Y'$ intersect at point $Z.$ Let lines $X'Y$ and $XY'$ intersect at point $Z'.$ Prove that $OZ$ and $OZ'$ are the isogonals with respect to the pair $(\ell,O).$

Proof

Let us perform a projective transformation of the plane that maps the point $O$ into a point at infinity and the line $\ell$ maps to itself. In this case, the isogonals turn into a pair of straight lines parallel to $\ell$ and equidistant from $\ell.$

The reverse (also projective) transformation maps the points equidistant from $\ell$ onto isogonals.

vladimir.shelomovskii@gmail.com, vvsss

Definition of isogonal conjugate of a point

Definitin 1.png

Let $P$ be a point in the plane, and let $ABC$ be a triangle. We will denote by $a,b,c$ the lines $BC, CA, AB$. Let $p_a, p_b, p_c$ denote the lines $PA$, $PB$, $PC$, respectively. Let $q_a$, $q_b$, $q_c$ be the reflections of $p_a$, $p_b$, $p_c$ over the angle bisectors of angles $A$, $B$, $C$, respectively. Then lines $q_a$, $q_b$, $q_c$ concur at a point $Q$, called the isogonal conjugate of $P$ with respect to triangle $ABC$.

Proof

By our constructions of the lines $q$, $\angle p_a b \equiv \angle q_a c$, and this statement remains true after permuting $a,b,c$. Therefore by the trigonometric form of Ceva's Theorem \[\frac{\sin \angle q_a b}{\sin \angle c q_a} \cdot \frac{\sin \angle q_b c}{\sin \angle a q_b} \cdot \frac{\sin \angle q_c a}{\sin \angle b q_c} = \frac{\sin \angle p_a c}{\sin \angle b p_a} \cdot \frac{\sin \angle p_b a}{\sin \angle c p_b} \cdot \frac{\sin \angle p_c b}{\sin \angle a p_c} = 1,\] so again by the trigonometric form of Ceva, the lines $q_a, q_b, q_c$ concur, as was to be proven. $\blacksquare$

Second definition

Definition 2.png

Let triangle $\triangle ABC$ be given. Let point $P$ lies in the plane of $\triangle ABC,$ \[P \notin AB, P \notin BC, P \notin AC.\] Let the reflections of $P$ in the sidelines $BC, CA, AB$ be $P_1, P_2, P_3.$

Then the circumcenter $Q$ of the $\triangle P_1P_2P_3$ is the isogonal conjugate of $P.$

Points $A, B,$ and $C$ have not isogonal conjugate points.

Another points of sidelines $BC, AC, AB$ have points $A, B, C,$ respectively as isogonal conjugate points.

Proof \[PC = P_1C, PC = P_2C \implies P_1C = P_2C.\] \[\angle ACQ = \angle BCP_1 \implies \angle QCP_1 = \angle ACB.\] \[\angle BCQ = \angle ACP_2 \implies \angle QCP_2 = \angle ACB.\] $\angle QCP_1 = \angle QCP_2, CP_1 = CP_2, QC$ common $\implies$ \[\triangle QCP_1 = \triangle QCP_2 \implies QP_1 = QP_2.\] Similarly $QP_1 = QP_3 \implies Q$ is the circumcenter of the $\triangle P_1P_2P_3.$ $\blacksquare$

From definition 1 we get that $P$ is the isogonal conjugate of $Q.$

It is clear that each point $P$ has the unique isogonal conjugate point.

Let point $P$ be the point with barycentric coordinates $(p : q : r),$ \[p = [(P-B),(P-C)], q = [(P-C),(P-A)], r = [(P-A),(P-B)].\] Then $Q$ has barycentric coordinates \[(p' : q' : r'), p' = \frac {|B - C|^2}{p}, q' = \frac {|A-C|^2}{q}, r' = \frac {|A - B|^2}{r}.\]

vladimir.shelomovskii@gmail.com, vvsss

Distance to the sides of the triangle

Distances to.png

Let $Q$ be the isogonal conjugate of a point $P$ with respect to a triangle $\triangle ABC.$

Let $E$ and $D$ be the projection $P$ on sides $AC$ and $BC,$ respectively.

Let $E'$ and $D'$ be the projection $Q$ on sides $AC$ and $BC,$ respectively.

Then $\frac {PE}{PD} = \frac{QD'}{QE'}.$

Proof

Let $\theta = \angle ACP = \angle BCQ, \Theta = \angle ACQ = \angle BCP.$ \[\frac {PE}{PD} = \frac {PC \sin \theta}{PC \sin \Theta} = \frac {QC \sin \theta}{QC \sin \Theta} = \frac {QD'}{QE'}.\] vladimir.shelomovskii@gmail.com, vvsss

Sign of isogonally conjugate points

Isog dist.png
Isog distance.png

Let triangle $\triangle ABC$ and points $P$ and $Q$ inside it be given.

Let $D, E, F$ be the projections $P$ on sides $BC, AC, AB,$ respectively.

Let $D', E', F'$ be the projections $Q$ on sides $BC, AC, AB,$ respectively.

Let $\frac {PE}{PD} = \frac{QD'}{QE'}, \frac {PF}{PD} = \frac{QD'}{QF'}.$ Prove that point $Q$ is the isogonal conjugate of a point $P$ with respect to a triangle $\triangle ABC.$

One can prove similar theorem in the case $P$ outside $\triangle ABC.$

Proof

\[\frac {PE}{PD} = \frac {PE}{PC} : \frac {PD}{PC} = \frac {\sin \angle ACP}{\sin \angle BCP},\] \[\frac {QD'}{QE'} = \frac {QD'}{QC} : \frac {QE'}{QC} = \frac {\sin \angle BCQ}{\sin \angle ACQ}.\]

Denote $\angle ACP = \varphi, \angle BCQ = \psi, \angle ACB = \gamma.$ \[\sin \varphi \cdot \sin (\gamma - \psi) = \sin \psi \cdot \sin (\gamma - \varphi) \implies\] \[\cos (\varphi - \gamma + \psi) - \cos(\varphi + \gamma - \psi) = \cos (\psi - \gamma + \varphi) - \cos(\psi + \gamma - \varphi)\] \[\cos (\gamma + \varphi -\psi) = \cos(\gamma - \psi + \varphi) \implies\] \[\cos \gamma \cos (\varphi - \psi) - \sin \gamma \sin (\varphi - \psi) = \cos \gamma \cos (\varphi - \psi) + \sin \gamma \sin (\varphi - \psi)\] \[2 \sin \gamma \cdot \sin (\varphi - \psi) = 0, \varphi + \psi < 180^\circ \implies \varphi = \psi.\] Similarly $\angle ABP = \angle CBQ \implies$ point $Q$ is the isogonal conjugate of a point $P$ with respect to a triangle $\triangle ABC.$

vladimir.shelomovskii@gmail.com, vvsss

Circumcircle of pedal triangles

Common circle.png

Let $Q$ be the isogonal conjugate of a point $P$ with respect to a triangle $\triangle ABC.$ Let $E, D, F$ be the projection $P$ on sides $AC, BC, AB,$ respectively.

Let $E', D', F'$ be the projection $Q$ on sides $AC, BC, AB,$ respectively.

Then points $D, D', E, E', F, F'$ are concyclic.

The midpoint $PQ$ is circumcenter of $DD'EE'FF'.$

Proof

Let $\theta = \angle ACP = \angle BCQ, \Theta = \angle ACQ = \angle BCP.$ $CE \cdot CE' = PC \cos \theta \cdot QC \cos \Theta = PC \cos \Theta \cdot QC \cos \theta = CD \cdot CD'.$ Hence points $D, D', E, E'$ are concyclic.

$PQE'E$ is trapezoid, $E'E \perp PE \implies OE = OE' \implies$

the midpoint $PQ$ is circumcenter of $DD'EE'.$

Similarly points $D, D', F, F'$ are concyclic and points $F, F', E, E'$ are concyclic.

Therefore points $D, D', E, E', F, F'$ are concyclic, so the midpoint $PQ$ is circumcenter of $DD'EE'FF'.$

vladimir.shelomovskii@gmail.com, vvsss

Common circumcircle of the pedal triangles as the sign of isogonally conjugate points

Let triangle $\triangle ABC$ and points $P$ and $Q$ inside it be given. Let $D, E, F$ be the projections $P$ on sides $BC, AC, AB,$ respectively. Let $D', E', F'$ be the projections $Q$ on sides $BC, AC, AB,$ respectively.

Let points $D, E, F, D', E', F'$ be concyclic and none of them lies on the sidelines of $\triangle ABC.$

Then point $Q$ is the isogonal conjugate of a point $P$ with respect to a triangle $\triangle ABC.$

This follows from the uniqueness of the conjugate point and the fact that the line intersects the circle in at most two points.

vladimir.shelomovskii@gmail.com, vvsss

Circles

2 points isogon.png

Let $Q$ be the isogonal conjugate of a point $P$ with respect to a triangle $\triangle ABC.$ Let $D$ be the circumcenter of $\triangle BCP.$ Let $E$ be the circumcenter of $\triangle BCQ.$ Prove that points $D$ and $E$ are inverses with respect to the circumcircle of $\triangle ABC.$

Proof

The circumcenter of $\triangle ABC$ point $O,$ and points $D$ and $E$ lies on the perpendicular bisector of $BC.$ \[\angle BOD = \angle COE = \angle BAC.\] \[2 \angle BDO = \angle BDC = \overset{\Large\frown} {BC} = 360^\circ - \overset{\Large\frown} {CB} = 360^\circ - 2 \angle BPC.\] \[\angle BDO = 180^\circ - \angle BPC = \angle PBC + \angle PCB.\] Similarly $\angle CEO = 180^\circ - \angle BQC = \angle QBC + \angle QCB.$ \[\angle PBC + \angle QBC = \angle PBC + \angle PBA = \angle ABC.\] \[\angle QCB + \angle PCB = \angle QCB + \angle QCA = \angle ACB.\]

\[\angle CEO +\angle BDO = \angle ABC + \angle ACB = 180^\circ - \angle BAC \implies\] \[\angle OBD = 180^\circ - \angle BOD - \angle BDO = \angle OEC \implies\] \[\triangle OBD \sim \triangle OEC \implies \frac {OB}{OE} = \frac {OD}{OC} \implies OD \cdot OE = OB^2. \blacksquare\]

vladimir.shelomovskii@gmail.com, vvsss

Problems

Olympiad

Given a nonisosceles, nonright triangle $ABC,$ let $O$ denote the center of its circumscribed circle, and let $A_1, \, B_1,$ and $C_1$ be the midpoints of sides $BC, \, CA,$ and $AB,$ respectively. Point $A_2$ is located on the ray $OA_1$ so that $\triangle OAA_1$ is similar to $\triangle OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1,$ respectively, are defined similarly. Prove that lines $AA_2, \, BB_2,$ and $CC_2$ are concurrent, i.e. these three lines intersect at a point. (Source)

Let $P$ be a given point inside quadrilateral $ABCD$. Points $Q_1$ and $Q_2$ are located within $ABCD$ such that $\angle Q_1 BC = \angle ABP$, $\angle Q_1 CB = \angle DCP$, $\angle Q_2 AD = \angle BAP$, $\angle Q_2 DA = \angle CDP$. Prove that $\overline{Q_1 Q_2} \parallel \overline{AB}$ if and only if $\overline{Q_1 Q_2} \parallel \overline{CD}$. (Source)

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