PROBLEM XVIII.

Given the trapezium ABCD, and a point E, in one of the sides, to find a point in each of the other sides, so that if an ellipsis was to be inscribed, it would touch the trapezium in these points.

1. Produce the sides of the trapezium, till they meet at K and L.

2. Draw the diagonals A C, and B D, cutting each other at F; produce B D, till it cut K L, at M.

3. Through F, and the given point E, draw E G, cutting BC at G.

4. From M, through the points E, and G, draw M H, and M G, cutting the other two sides in the points I and H, then E, H, G, I, will be the four points required.

PROBLEM XIX.

A trapezium ABCD being given, and a point E, in one of the sides, to find the centre of an ellipsis that may be described in the trapezium, and pass through the point of contact È, without drawing any part of the ellipsis.

1. Find the points of contact H, G, I, E, as in the last problem.

2. Join the points G, and E, by the right line G E; bisect it in M, and from K, where the opposite sides A D, and B C meet, and through the point M, draw K M indefinitely.

3. Also join any other two points of contact, as HI; bisect H I, at N, from L, where the opposite sides B A, and C D meet; draw L N, meeting K M, at P, then P will be the centre of the ellipsis required.

And in like manner if the points G, and H, were joined, and bisected at Q, and a line being drawn from B, where the opposite sides A B, and C D meet through Q, it would also meet in P, the centre, &c.

PROBLEM

PROBLEM XX.

Given a trapezium ABC D, and a point E, in one of the sides, to find the two axes of an ellipsis that may be inscribed in the trapezium, and pass through the point E, without drawing any part of the ellipsis.

1. Find the opposite points of contact H, E, F, G, by problem XVIII.

2. From thence, find the centre P, by the last problem.

3. From E, and through the centre P, draw E M, making P M equal to P E.

4. Through H, or any other point of contact, draw HK, parallel to D C, cutting E G`at K; then K H is an ornate to the diameter E M.

5. Through P, the centre, draw PR parallel to H K.

6. Find the extremities R and S, of the diameter R S, by problem XI.

7. The conjugate diameters E M, and R S, being now found, then find the two axes V W, and X Y, by problem XIII.