Brianchon’s Theorem

When you rotate a straight line about the vertical axis, you will generally get a hyperboloid of revolution. By construction, this is a ruled surface, and by symmetry, there is a second set of lines on the surface. We call these two sets of lines the A-lines and B-lines.

These lines dissect the hyperboloid into lots of skew quadrilaterals, reminding us that any quadrilateral can be doubly ruled, and opening up more possibilities for our previously discussed bent rhombi.

Let’s form a hexagon, following the A- and B-lines alternatingly once around the hyperboloid. Then a theorem by Charles Julien Brianchon states that the three main diagonals (i.e. those connecting opposite vertices) of this hexagon will meet in one point.

One reason why this is curious is that it quite unexpected: In space, we don’t even expect two lines to meet in a point, let alone three. The other reason is that it has such a simple proof, due to Germinal Pierre Dandelin: Any pair of A- and B-lines will lie in a common plane, because they either intersect or are parallel. So the pairs of opposite edges give us three planes, which will meet in a common point. Because the diagonals of the hexagon are also the intersections of any pair of the three planes, we are done.

If we project the hyperboloid with all its decorations into the plane, like done so in the images above, the outline of the hyperboloid becomes a common hyperbola, and the six lines of the hexagon tangential to it. This leads to Brianchon’s theorem in the plane: The main diagonals of a hexagon circumscribed in a conic section meet in a point.

This theorem becomes easier to parse if the conic is just an ellipse:

We also have enough room here to see that there is a second dual conic on which the A-lines and B-lines, respectively, meet.