Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/333

Rh Describe a circle to pass through A and B and to touch the circumference considered (10): the point of contact will be the required point. The demonstration is similar to that in the preceding proposition.

22. A and B are two given points without a given circle; it is required to determine the points on the circumference of the given circle at which AB subtends the greatest and least angles.

Suppose that neither AB nor AB produced cuts the given circle.

Describe two circles to pass through A and B, and to touch the given circle (10): the point of contact of the circle which touches the given circle externally will be the point where the angle is greatest, and the point of contact of the circle which touches the given circle internally will be the point where the angle is least. The demonstration is similar to that in 20.

If AB cuts the given circle, both the circles obtained by 10 touch the given circle internally; in this case the angle subtended by AB at a point of contact is less than the angle subtended at any other point of the circumference of the given circle which is on the same side of AB. Here the angle is greatest at the points where AB cuts the circle, and is there equal to two right angles.

If AB produced cuts the given circle, both the circles obtained by 10 touch the given circle externally; in this case the angle subtended by AB at a point of contact is greater than the angle subtended at any other point of the circumference of the given circle which is on the same side of AB. Here the angle is least at the points where AB produced cuts the circle, and is there zero.