Bisect DG at B, then about D describe an arc with any radius DQ greater than DB, and about O another are with radius OQ = DQ-FO; draw from Q the intersections of these arcs, the line QD, and also QO, producing the latter to cut the circumference in E. By this process we may construct the curve QBZ, each point of which is also equally distant from the given point D, and from the concave instead of the convex arc of the given circumference.  The difference between QD and QO being constant and equal to FO, and AB being also equal to FO, this curve is the other branch of the same hyperbola, whose major axis is equal to the radius of the given circle.

The tangent at P bisects the angle DPL, and is perpendicular to DL, which it bisects at a point I on the circumference of the circle whose diameter is AB, the major axis, the center being C, the middle point of D O. As P recedes from A, it is evident that the angles P D L, P L D, will increase, until D L assumes the position D T tangent to the given circle, when they will become right angles.  P will therefore be infinitely remote, and the point I having then reached t, where D T touches the smaller circle, C t S will be an asymptote to the curve.  This shows that the measurements from the convex arc, for the construction of A P, are made only from the portion F T of the given circumference.

In the diagram the point Q is so chosen that D L produced passes through E, so that Q J, the tangent at Q, is parallel to P I. It will thus be seen that the measurements from the concave arc, for the construction of B Q, are confined to the portion G T of the given circumference.  As D L E rises, the points P and Q recede from A and B, the points L and E approach each other, finally coinciding at T; at this instant I and J fall together at t, so that S S is the common asymptote to A P and B Q.

In Fig. 2 the given point D lies within the circumference of the given circle.  Bisect D F at A, and D G at B; about D describe an arc with any radius D P greater than D A, and about O another, with radius O P = O F—­D P, these arcs intersect in P, and producing O P to cut the circumference in L, we have P D = P L. Similarly E D = E H, U D = U W, etc.  And since P D + P O = L P + P O, D E + E O = H E + E O, and so on, the curve is obviously the ellipse of which the foci are D and O, and the major axis is A B = F O, the radius of the given circle.

[Illustration:  FIG 2.]

If, as in Fig. 3, the given point be made to coincide with the center of the circle, the ellipse becomes a circle with diameter A B = F O. But if the point be placed upon the circumference, as in Fig. 4, the ellipse will reduce to the right line A B coinciding with F O.

[Illustration:  FIGS 3, 4, 5, 6.]

In this case we may also apply the same process as in Fig. 1; D T becomes a tangent at D to the circumference, and the asymptotes coincide with the axis of the hyperbola, of which one branch reduces to the right line A P extending from A to infinity on the left, and the other reduces to the right line B G Q, extending from B to infinity on the right.