Pascal's Theorem on the circle
Back to Geometry homepage
The picture above has a circle with centre O and a point P on the circle. The points A, B, C, D, E and F are all free to move around the circle.
Let X be the intersection of AE and BD, let Y be the intersection of AF and CD and let Z be the intersection of BF and CE.
Pascal's theorem then guarantees that the intersection points X, Y and Z are collinear.
You can click and drag the points O and P to move the circle, and you can click and drag the points A, B, C, D, E, F to move them around the circle. As you can see, the intersection points X, Y, Z remain collinear.
Pascal's theorem is an example of a theorem in projective geometry. If we apply a projective transformation to the plane, the circle becomes a conic section and properties such as intersection and collinearity are preserved. Therefore Pascal's theorem also applies to six points on a conic section.
