Geometry • Topic 33

Poles and Polars

This topic connects points and lines via a circle. It provides a geometric way to "reciprocate" problems (swapping points for lines).

Definition

Definition (Pole and Polar)

Given a circle

ω

(center

O

, radius

r

) and a point

P

(the pole): The polar of

P

is a line

ℓ

perpendicular to

OP

at a point

X

such that:

OP \cdot OX = r^{2}

If $P$ is outside $ω$ , the polar $ℓ$ passes through the points of tangency from $P$ .
If $P$ is on $ω$ , the polar is the tangent at $P$ .

La Hire's Theorem (Reciprocity)

Theorem (La Hire's Theorem)

Point

A

lies on the polar of

B

if and only if point

B

lies on the polar of

A

Brocard's Theorem

In a cyclic quadrilateral inscribed in circle

ω

with center

O

Intersection of sides $A B \cap C D = P$ .
Intersection of sides $A D \cap BC = Q$ .
Intersection of diagonals $A C \cap B D = R$ .

Then

R

is the orthocenter of

△ OPQ

, and

PQ

is the polar of

R

Practice Problems

Exercise (Problem 1)

Prove that the polar of the orthocenter of a triangle with respect to its polar circle is the line at infinity? (Advanced). Simpler: Find the polar of

(2, 0)

with respect to

x^{2} + y^{2} = 1

Exercise (Problem 2)

Use poles and polars to prove that if a quadrilateral has an inscribed circle, the diagonals and the lines connecting contact points are concurrent.

Exercise (Problem 3)

Construct the polar of a point

P

inside a circle using only a straightedge.