For APN, APO are isosceles triangles, and give the angle PAN= PNA, and angle PAO = POA ... NPA' = 2PAN, OPA' = 2PAO; or, by addition, NPO 2NAO, i. e., NPO = 2BAO.

Similarly, the two triangles aPN, aPO give

NPO = 2NaO, or NPO = 2ba0 .. BAO = baO.

Reasoning upon the four points Q, B, b, N in the same manner as upon the four points P, A, a, N, it may be proved that the angles ABO and abO are equal. Thus the triangles OAB, Oab are similar, and give

OA: Oa:: OB: Ob:: AB: ab.

The same would result from any number of triangles.

Scholium. The point O is the only common homologous point of the two polygons. Every line passing through it is called a common homologous line, and conversely.

The center of similitude of two polygons is their common homologous point.

There is another mode of construction which seems more natural than that just given.

Determine the circumference of a circle, every point of which shall be at distances from two homologous vertices A, a in the ratio of similitude of the two polygons (see Prob. 12 of General Note, p. 95); repeat the same construction for two other vertices B, b; one of the points in which the two circumferences intersect will be the point sought.

CENTERS OF SIMILITUDE IN CIRCLES.

THEOREM.

Two circles, as well as two regular polygons of an even number of sides, have two centers of similitude, the one internal and the other external.

When the two circles are exterior to each other, prove that the points in which their common tangents meet are centers of similitude. This point may be found by dividing the line, joining the centers in the ratio of the radii.

When the circles touch each other externally, prove that the point of contact is an internal center of similitude; and that if they touch each other internally, the point of contact is an external center of similitude.

There exist several other remarkable particular cases.

For two concentric circles, the centers of similitude unite in the common center.

For two equal circles, the internal center of similitude is at the

middle of the line joining their centers; the external is at an infinite distance.

When one of the circles degenerates into a right line, 1°. The centers of similitude are, at the extremities of a diameter of the other circle, perpendicular to the line. 2°. If one of the circles reduces to a point, that point is itself the center of similitude, both internal and external.

Scholium. To be proved. When three circles are situated upon the same plane which gives six centers of similitude; 1o. The three external centers of similitude; 2°. One external and two internal—are upon a same line, which gives four lines, passing through six points combined, three and three.

RADICAL AXIS AND RADICAL CENTER.

Definitions. A radical axis of two circles is the locus of points from each of which equal tangents can be drawn to the two circles.

Construction. Divide the line joining the centers of the two circles in such a manner that the difference of the squares of the two parts is equal to the difference of the squares of the radii, and the perpendicular to this line at the point of division will be a radical axis.

Prove that to find the point on the line joining the centers it is only necessary to lay off from the middle of this line, on the side toward the smaller circle, a distance equal to half a third proportional to the distance between the centers and the square root of the difference of the squares of the radii. (See Prob. 14, p. 80.)

Each particular case, however, presents a more simple construction. 1o. If the circles be exterior, or in any position for which there exists a common tangent, as the middle point of the portion of this tangent comprehended between the two points of contact belongs to the radical axis, we draw through this point a perpendicular to the line joining the centers of the circles, and thus have this axis. 20. When the circles touch either exteriorly or interiorly, the common point of the two circumferences belongs necessarily to the radical axis, and thus leads to its determination, as before. It is then the common tangent to the circles at this point. 3°. When the circles cut each other, the common chord produced both ways is the radical axis.

Concentric circles have no radical axis. When the two circles are equal, the radical axis is the perpendicular at the middle of the line joining the centers.

If one of the circles be reduced to a point, the radical axis is obtained by joining the middles of the tangents drawn from this point to the other circle.

If one of the circles degenerates into a right line, the radical axis is the line itself.

RADICAL CENTER.-Three circles situated in the same plane (the centers of which are not in the same line) give, by their combination two and two, three radical axes; and these three axes cut each other in the same point.

For the two first cutting each other, and being respectively perpendicular to two lines which cut, their point of intersection is such that there can be drawn from this point to the three circumferences equal tangents; consequently, it belongs to the third radical axis.

This point is called the radical center of the three circles.

From this definition, and from what has been shown above, it follows, that if three circumferences intersect, the three chords which unite their points of intersection meet in the same point.

When this point of intersection is exterior to the three circles, the six tangents from this point are equal.

THEOREM. Prove that if, from any point of a radical axis of two circles, a secant be drawn meeting the circumferences in FOUR points, these four points will be in the circumference of a third circle.

This may be proved by aid of the theorem that the rectangle of a secant and its external segment is constant (th. 42), together with the construction of a radical axis.

THEOREM. If through one of two centers of similitude (external or internal) of two circles two secants to these circles be drawn, 10. The eight points of intersection combined, four and four, in a suitable manner, form FOUR groups, situated respectively upon as many new circumferences; 2°. These four circumferences have for a cOMMON RADICAL CENTER that center of similitude which served to determine these circumferences. Note. The above theory will be found of great use in the solution of all problems involving the contact of circles.

CONJUGATE POINTS, POLES, AND POLAR LINES. Conjugate points are two points situated the one within, the other without, a circle, in such a manner that the distances of every point in the circumference from these two points are in a constant ratio. The circle is called the regulating circle.

The point within the circle being given, to determine its conjugate, erect at the given point a perpendicular to the line joining the given point and the center, and at the point where this perpendicular meets the circumference draw a tangent which will meet the line joining the given point and center produced in the point required. Prove this.

A chord of contact is a line joining the points of contact of two tangents drawn from the same point.

THEOREM. The chords of contact of all tangents which meet in one and the same line will meet in the same point, and the conjugate of this point is the foot of a perpendicular let fall from the center of the circle upon the line in which the tangents meet.

The point in which all the chords of contact meet is called a pole, and the line in which the tangents all meet, a polar line.