# Pole. Polar. Theorems of Brianchon and Brokard

Definition

Given a circle $$k(O,r)$$, let $$A^*$$ be the image of the point $$A\neq O$$ under the inversion with respect to $$k$$. The line $$a$$ passing through $$A^*$$ and perpendicular to $$OA$$ is called the polar of $$A$$ with respect to $$k$$. Conversely $$A$$ is called the pole of $$a$$ with respect to $$k$$.

Theorem 6

Given a circle $$k(O,r)$$, let and $$a$$ and $$b$$ be the polars of $$A$$ and $$B$$ with respect to $$k$$. The $$A\in b$$ if and only if $$B\in a$$.

Definition

Points $$A$$ and $$B$$ are called conjugated with respect to the circle $$k$$ if one of them lies on a polar of the other.

Theorem 7

If the line determined by two conjugated points $$A$$ and $$B$$ intersects $$k(O,r)$$ at $$C$$ and $$D$$, then $$\mathcal H(A,B;C,D)$$. Conversely if $$\mathcal H(A,B;C,D)$$, where $$C,D\in k$$ then $$A$$ and $$B$$ are conjugated with respect to $$k$$.

Theorem 8 (Brianchon)

Assume that the hexagon $$A_1A_2A_3A_4A_5A_6$$ is circumscribed about the circle $$k$$. The lines $$A_1A_4$$, $$A_2A_5$$, and $$A_3A_6$$ intersect at a point.

From the previous proof we see that the Brianchon$$\prime$$s theorem is obtained from the Pascal$$\prime$$s by replacing all the points with their polars and all lines by theirs poles.

Theorem 9 (Brokard)

The quadrilateral $$ABCD$$ is inscribed in the circle $$k$$ with center $$O$$. Let $$E=AB\cap CD$$, $$F=AD\cap BC$$, $$G=AC\cap BD$$. Then $$O$$ is the orthocenter of the triangle $$EFG$$.

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