Brianchon's theorem

In geometry, Brianchon's theorem is a theorem stating that when a hexagon is circumscribed around a conic section, its principal diagonals (those connecting opposite vertices) meet in a single point.

be a hexagon formed by six tangent lines of a conic section.

(extended diagonals each connecting opposite vertices) intersect at a single point

This procedure results in a statement on inellipses of triangles.

That means there exists a central collineation, which maps the one onto the other triangle.

But only in special cases this collineation is an affine scaling.

For example for a Steiner inellipse, where the Brianchon point is the centroid.

Brianchon's theorem can be proved by the idea of radical axis or reciprocation.

To prove it take an arbitrary length (MN) and carry it on the tangents starting from the contact points: PL = RJ = QH = MN etc.

Draw circles a, b, c tangent to opposite sides of the hexagon at the created points (H,W), (J,V) and (L,Y) respectively.

One sees easily that the concurring lines coincide with the radical axes ab, bc, ca resepectively, of the three circles taken in pairs.

The theorem takes particular forms in the case of circumscriptible pentagons e.g. when R and Q tend to coincide with F, a case where AFE is transformed to the tangent at F. Then, taking a further similar identification of points T,C and U, we obtain a corresponding theorem for quadrangles.