polygon similar to the original, and so placed that the homologous sides of the two polygons are parallel.

282°. When two similar polygons are so placed as to have their homologous sides parallel, they are in perspective, and the joins of corresponding vertices concur at a centre of similitude.

Let ABCD..., abcd... be the polygons.

Since they are similar, AB : ab=BC : bc=CD : cd... (207°), and by hypothesis AB is || to ab, BC to bc, etc.

Let Aa and Bb meet at some point O.

Then OAB is a ▲ and ab is || to AB.

.. Cc passes through O, and similarly Dd passes through O,

etc.

By writing a'b'c'... for abc... the theorem is proved for the polygon a'b'c'd', which is oppositely placed to ABCD...

Cor. 1. If Aa and Bb meet at ∞, ab=AB, and hence bc=BC, etc., and the polygons are congruent.

Cor. 2. The joins of any two corresponding vertices as A, C ; a, c; a', c' are evidently homologous lines in the polygons and are parallel.

Similarly any line through the centre O, as XxOx' is homologous for the polygons and divides them similarly.

283°. Let the polygon ABCD... have its sides indefinitely increased in number and diminished in length. Its limiting form (148°) is some curve upon which its vertices lie. A similar curve is the limiting form of the polygons abcd..... as also of a'b'c'd'......., since every corresponding pair of limiting or vanishing elements are similar.

Hence, if two points on a variable radius vector have the ratio of their distances from the pole constant, the loci of the

points are similar curves in perspective, and having the pole as a centre of perspective or similitude.

Cor. 1. In the limiting form of the polygons, the line BC becomes a tangent at B, and the line bc becomes a tangent at b. And similarly for the line bc'.

.. the tangents at homologous points on any two curves in perspective are parallel.

284°. Since abcd..... and a'b'c'd'... are both in perspective with ABCD... and similar to it, we see that two similar polygons may be placed in two different relative positions so as to be in perspective, that is, they may be similarly placed or oppositely placed.

In a regular polygon of an even number of sides no distinction can be made between these two positions; or, two similar regular polygons are both similarly and oppositely placed at the same time when so placed as to be in perspective.

Hence two regular polygons of an even number of sides. and of the same species, when so placed as to have their sides respectively parallel, have two centres of perspective, one due to the polygons being similarly placed, the external centre ; and the other due to the polygons being oppositely placed, the internal centre.

Cor. Since the limiting form of a regular polygon is a circle (148°), two circles are always similarly and oppositely placed at the same time, and accordingly have always two centres of perspective or similitude.

285°. Let S and S' be two circles with centres C, C' and radii r, respectively, and let O and O' be their centres of perspective or similitude,

Let a secant line through O cut S in X and Y, and S' in X' and Y'.

Then O is the centre of similitude due to considering the

circles S and S' as being similarly placed.

Hence X and X', as also Y and Y', are homologous points, and (283°, Cor. 1) the tangents at X and X' are parallel. So also the tangents at Y and Y' are parallel.

Again O' is the centre of similitude due to considering the circles as

being oppositely placed,

S

and for this centre Z and Y' as also U and U' are homologous points; and tangents at Y and Z are parallel, and so also are tangents at U and U'.

Hence YZ is a diameter of the circle S and is parallel to Y'Z' a diameter of the circle S'.

Hence to find the centres of similitude of two given circles :-Draw parallel diameters, one to each circle, and connect their end-points directly and transversely. The direct connector cuts the common centre-line in the external

Q

centre of similitude, and the transverse connector cuts it in the internal centre of similitude.

286°. Since OX: OX'=OY: OY', if X and Y become coincident, X' and Y' become coincident also.

.. a line through O tangent to one of the circles is tangent to the other also, or O is the point where a common tangent cuts the common centre-line. A similar remark applies to O'.

When the circles exclude one another the centres of similitude are the intersections of common tangents of the same name, direct and transverse.

When one circle lies within the other (2nd Fig.) the common tangents are imaginary, although O and O' their points of intersection are real.

287°. Since AOCY≈▲OC'Y', .. OC:OC'=r: r', and since ▲O'CZ≈▲O'C'Y', .. O'C: O'C'=r: r'. .. the centres of similitude of two circles are the points which divide, externally and internally, the join of the centres of the circles into parts which are as the conterminous radii. The preceding relations give

I. O lies within the circle S when the distance between the centres is less than the difference of the radii, and O' lies within the circle S when the difference between the centres is less than the sum of the radii.

2. When the circles exclude each other without contact both centres of similitude lie without both circles.

3. When the circles touch externally, the point of contact is the internal centre of similitude.

4. When one circle touches the other internally, the point of contact is the external centre of similitude.

5. When the circles are concentric, the centres of similitude coincide with the common centre of the circles, unless the circles are also equal, when one centre of similitude becomes any point whatever.

6. If one of the circles becomes a point, both centres of similitude coincide with the point.

288°. Def.--The circle having the centres of similitude of two given circles as end-points of a diameter is called the circle of similitude of the given circles.

The contraction of s. will be used for circle of similitude.

P

Z

Cor. 1. Let S, S' be two circles and Z their ○ of s. Since O and O' are two points from which tangents to circles S and S' are in the constant ratio of r to r', the circle o Z is co-axal with S and

S' (275°, Cor. 5). Hence

T

T

C

C'

S

s'

any two circles and their of s. are co-axal.

Cor. 2. From any point P on circle Z,

Hence, at any point on the of s. of two circles, the two circles subtend equal angles.

Cor. 3.

whence

I.

.. 1. The of s. is a line, the radical axis, when the given circles are equal (r=r').

2. The of s. becomes a point when one of the two given. circles becomes a point (r or r'=0).

3. The of s. is a point when the given circles are concentric (CC'=0).