Villarceau showed that given an arbitrary point on a torus, four of these can be drawn passing through that point.
Clifford's theorem is be used to show that an infinite set of these with common intersection points can be constructed given four of these which intersect at one point. Villarceau showed that given an arbitrary point on a torus, four of these can be drawn passing through that point. Apollonius defined these objects as the set of points which have a constant ratio for distances to two fixed points, while Feuerbach's theorem describes one which passes through nine important points on a triangle. Represented by the parametric equations x = a cos t and y = a sin t in Cartesian space, for 10 points, identify these geometric structures whose circumference is given by the formula 2 pi R.
Clifford's theorem is be used to show that an infinite set of these with common intersection points can be constructed given four of these which intersect at one point.
{circles} [do not accept or prompt on ellipses]