The Elements

If, therefore, the surface of a cone is cut by that through which the plane is written, making the circle obliquely, the section will be what is called the circumference of an ellipsis ellipse; and let a certain straight line be drawn through the center of the circle perpendicular to the diameter bg and to the plane through its axis.
of it

And let the straight lines ab, ag, and ad be drawn from a to the bg; and from the vertex a to the base bg, the straight lines ae and az, which, having been drawn from the center toward the circumference, are equal to each other; but those not drawn toward the sections are unequal to each other; and each of those drawn toward the sections is greater than any of those not drawn toward the sections.

A small triangle with vertex A and base BC. A horizontal line segment intersects the triangle, labeled with Greek letters.

from

For let there be a cone whose vertex is a, and its base is the circle bg; and let it be cut by a plane through the axis; and let it make in its surface the straight lines ab and ag, and in the base the diameter bg; and let a certain straight line ad be drawn in the triangle abg from the vertex a to the base bg; and from the vertex a toward the circumference of the circle, the straight lines ae and az.
A triangle with vertex at top and base at bottom, showing a vertical line from the vertex to the base and an additional line extending towards the right.
I say that the straight line ad is greater than the straight line ae; and ae [is greater] than az. For let them be drawn from the center of the circle toward the sections of the circumference...

Of all the lines drawn from the center to the circumference, those drawn toward the sections of the circular circumferences of the base of the cone are equal to each other; but those not drawn toward the sections are unequal to each other; and each of those drawn toward the sections is greater than any of those not drawn toward the sections; and the one nearer to the one drawn toward the sections is greater than the one farther away; and those equidistant from each of the lines drawn toward the sections are equal to each other.

from the centers

It is clear that of the straight lines led toward the base of the triangle, those toward the sections are equal; but those toward the others are unequal; and the one nearer to the one drawn toward the sections is greater than the one farther away; and those equidistant are equal; and of the straight lines led toward the base, the same [ones] are greater [than] the straight lines led toward the base at right angles.

the angles of the triangles

A large triangle with vertex labeled alpha and base labeled beta and gamma. A central vertical line drops from alpha to point delta. Two additional lines from the vertex alpha connect to points on the base line beta-gamma. To the right of this diagram is a smaller triangle with internal lines, and between them is an oval institutional library stamp.

Let there be a cone whose vertex is a, and whose base is the circle bg; and let it be cut by a plane through the axis; and let it make in its surface the straight lines ab and ag, and in the base the diameter bg; and let a certain straight line ad be drawn in the triangle abg from the vertex a to the base bg.

Two small triangles side-by-side at the bottom, illustrating geometric proportions.