Poles and Polars

billzhao

I've been seeing a lot of that on the forum lately. I think I get what it is, but could someone tell me where and how it is generally used in Olympiad problems? And where did you guys learn it?

Also, is "polar" geometry is part of projective geometry or inversive geometry?

**Posted:** Mon May 03, 2004 8:00 am

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Yufei Zhao

Valentin Vornicu

**Posted:** Mon May 03, 2004 9:12 am

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grobber

We are given a circle $(O)$ and a point $P$ . We draw a chord of the circle through $P$ , cutting the circle at $A$ and $B$ . The locus of point $X$ s.t. $(A,B;P,X)=-1$ is a line which is obviously (for symmetry reasons) perpendicular to $OP$ (O is the center of the circle $(O)$ ). This is called the circular polar of $P$ wrt $(O)$ .

Let $\angle xOy$ be an angle and let $P$ be a point in the plane. Draw a line through $P$ which cuts $Ox$ and $Oy$ in $A$ and $B$ respectively. The locus of $X$ s.t. $(A,B;P,X)=-1$ is a line passing through $O$ . This is called the angular polar of $P$ wrt $\angle xOy$ .

A very useful fact: Let $A$ and $B$ be two arbitrary points (different from the center of the circle $(O),\ O$ . Then $A$ is on the circular polar of $B$ iff $B$ is on the circular polar of $A$ (something similar can be done for angular polars, of course).

It's part of projective geometry. Here's a connection between the angular polar and the circular polar:

Let $ABCD$ be a quadrilateral inscribed in $(O)$ . $E=AB\cap CD,\ F=AD\cap BC$ . Then the angular polar of $E$ wrt $\angle AFB$ = the circular polar of $E$ wrt $(O)$ .

**Posted:** Tue May 04, 2004 3:20 am

billzhao

I don't quite understand your notation. What does $(A,B;P,X)=-1$ mean?

**Posted:** Wed May 05, 2004 7:56 am

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grobber

$(A,B;C,D)$ means cross-ratio and it's equal to $\frac{CA}{CD}$ : $\frac{DA}{DB}$ , where all the 2-letter combinations are oriented segments. The four points must be collinear, of course.

If we have four fixed lines and we cut them with a variable line, then the cross-ratio of the four intersection points doesn't change, so we can also speak about the cross-ratio of four lines.

**Posted:** Wed May 05, 2004 8:06 am

Charlesww · New Member Offline Joined: 29 Sep 2004 Posts: 12

See also http://www.cut-the-knot.org/Curriculum/Geometry/PolePolar.shtml for general facts regarding poles and polars.

**Posted:** Sat Oct 09, 2004 8:22 am

Altheman

**Posted:** Fri Jun 15, 2007 4:03 pm

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Altheman's Problem Column

Virgil Nicula

Poles and polars come in pairs. Poles are plane points; polars are straight lines in the same plane. Below, I define the pole/polar correspondence with respect to a circle.

So let there be a circle $w=\mathcal R (O,R)$ . Let $A$ be an arbitrary point, except $O$ , whose inverse image in $w$ is $A'$ , i.e. $OA\cdot OA' = R^{2}$ . The point $A'\in OA$ . Let $a$ be the line through $A'$ perpendicular to $OA'$ (or $OA$ , which is the same.) Then $a$ is called the polar of $A$ , while $A$ is the pole of $a$ w.r.t. to $w$ .

If $A$ is outside $w$ , we may draw two tangents from $A$ to $w$ . Then the polar $a$ is the line through the two points of tangency. If $A$ lies on $w$ , its polar $a$ is just the tangent to $w$ at $A$ .

Buy this applet

The applet allows adding any number of pole/polar pairs. This can be done in two ways. You can add a pole which will appear as a small hollow circle that could be dragged. Its polar will be drawn automatically. It is also possible to add a polar - a straight (draggable) line defined by two draggable points. Its polar will then appear as a small filled circle.

Observe the main properties of poles and their polars:

The polar of a pole that lies inside the circle of reference lies in its entirety outsdie the circle.

The polar of a pole on the circle is tangent to the circle at the pole.

The polar of a pole that lies outside the circle of reference crosses the circle at two points. The lines joining the points to the pole are tangent to the circle.

If point $A$ lies on the polar of point $B$ , then point $B$ lies on the polar of point $A$ . (La Hire's Theorem.)

The pole of a line through two poles lies at the intersection of their polars.

The polar of a point which is the intersection of two polars, is the line through the corresponding poles.

The polars of three collinear points are concurrent.

The poles of three concurrent lines are collinear.

The latter properties are a direct consequence of La Hire's theorem. For example, assume points $A,\ B,\ C$ with polars $a,\ b,\ c$ respectively, lie on a straight line $m$ . Let $M$ be the pole of $m$ . Since each of $A,\ B,\ C$ lies on $m$ , each of $a,\ b,\ c$ passes through $M$ . They are therefore concurrent. Conversely, let three lines $a,\ b,\ c$ concur at point $M$ . Then each of $A,\ B,\ C$ lies on the polar $m$ of $M$ . They are therefore collinear.

Note that one needs only a straightedge to construct the polar of a point. That tangents to a circle from a point are straightedge constructible even in the absence of the center of the circle. (Compare that to Steiner's theorem.)

Let $A'$ be the inverse image of point $A$ lying outside the circle of inversion. $A'$ then lies inside the circle and, by definition, $OA\cdot OA' = R^{2}\ \ (1)$ .

Find points $S$ and $T$ on the circle such that $ST$ is perpendicular to $OA$ at $A'$ . From $(1)$ and the fact that they share an angle at $O$ , triangles $OAT$ and $OTA'$ are similar $(SAS)$ . Since angle $OA'T$ is right, so is angle $OTA$ . In other words, $AT$ is perpendicular to the radius $OT$ and is therefore tangent to the circle at $T$ .

**Posted:** Thu Jul 26, 2007 9:18 pm