If a scalene cone ADBV (fig. 44) be cut through the axis by a plane perpendicular to the base, making the triangle VAB, and from any point II in the straight line. AV a straight line IIK be drawn in the plane of the triangle VAB, so that the angle YE K may be equal to the angle VBA, and the cone be cnt by another plane passing through HK perpendicular to the plane of the triangle ABV, the common section HFKN of this plane and the cone will be a circle.
Take any point L in the straight line IIK, and through L draw EG parallel to AB, and let EFGN be a section parallel to the base, passing through EG ; then the two planes IIFKN, EFGN being perpendicular to the plane VAB, their common section FLN is perpendicular to ELG, and since EFGN is a circle (by last Prop.), and EG its diameter, the square of FL is equal to the rectangle contained by EL and LG (Fuel. iii. 35); but since the angle. VHK is equal to VBA or VGE, the angles EIIK,EGK are equal, therefore the points E, II, G, K, are it. the circumference of a circle (Eucl. iii. 211, and HL. =EL .LG (Eucl. iii. 35) =FL', therefore the section IIFKN is a circle of which HLK is a diameter (Encl. iii. 35).
This section is called a Snbnontrary Section.
If a cone be cut by a plane which does not pass through the vertex, and which is neither parallel to the base ncr to the plane of a subeontrary section, the common section of the plane and the surface of the cone will be an ellipse, a parabola, or an hyperbola, according as the _plane passing through the vertex parallel to the cutting plane falls without the cone, touches it, or falls within it.
Let ADBV (figs. 45, 46, 47) be any cone, and let ONP be the common section of a plane passing through its vertex and the plane of the base, which will either fall without the base, or touch it, or fall within it.
Let FKM be a section of the cone parallel to VP0 ; through C the centre of the base draw ON perpendicu lar to OP, meeting the circumference of the base in A and B ; let a plane pass through V, A, and B, meeting the plane OVP in the line NV, the surface of the cone in VA, VB, and the plane of the section FKM in g LK ; then, because the planes OVP, MK are parallel, KL will be parallel to VN, and will meet VB one side of the cone in K; it will either meet VA the other side in H, as in fig. 45, within the cone ; or it will be parallel to VA, as in fig. 46 ; or it will meet VA, produced beyond the vertex, in H, as in fig. 47.
Let EFGM be a sec tion of the cone parallel to the base, meeting the plane VAB in EG, and the plane FKM in FM, and let L be the intersection of EG and FM ; then EG will lie parallel to NB, and FM will he parallel to P0, and therefore will make the same angle with LK, wherever the lines FM,. LK cut each other; and since I3N is perpendicular to P0, EG is perpendicular to FM. Now the section EFGM is a circle of which EG is the diameter (Prop. ii.), therefore FM is bisected at L, and FL2= EL . LG.