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  "body": "\n## Summary\n\nCut a right circular cone by a plane.\nThe Dandelin spheres of this plane section are the spheres that touch the cone along a circle and also touch the cutting plane.\nFor an ellipse there are two, and their points of contact with the plane are the two foci of the ellipse.\nThey give a short proof that the plane sections of a cone are the curves defined by foci and distances.\nSo the two classical definitions of the [[conic sections|00HJCMC2CD]] agree.\nThey are named after Germinal Pierre Dandelin.\n\n## Explanations\n\n### L1 human\n\nSlice an ice-cream cone with a tilted plane: the cut is an ellipse.\nNow drop one ball into the cone above the plane and one below it.\nEach ball sits snugly against the cone along a circle and just touches the plane.\nThe two balls touch the plane at two points, and these points are exactly the foci of the ellipse.\nThe reason is that the two tangent segments from any point to a ball have the same length.\nSo the sum of distances to the foci equals the distance between the two circles along the cone.\nThis distance is the same for every point of the ellipse.\n",
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