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ON CENTRES OF SIMILARITY AND SIMILITUDE.
1. If any two unequal similar figures are placed so that their homologous sides are parallel, the lines joining corresponding points in the two figures meet in a point, whose distances from any two corresponding points are in the ratio of any pair of homologous sides.
Let ABCD, A'B'C'D' be two similar figures, and let them be placed so that their homologous sides are parallel ; namely, AB, BC, CD, DA parallel to A'B', B'C', C'D', D'A' respectively. Then shall AA', BB', CC', DD' meet in a point, whose distances from
any two corresponding points shall be in the ratio of any pair of homologous sides. Let AA' meet BB', produced if necessary, in S. Then because AB is par to A'B';
Нур. :: the As SAB, SA'B' are equiangular ;
: SA : SA' =AB : A'B'; :. AA' divides BB', externally or internally, in the ratio of AB to A'B'.
Similarly it may be shewn that CC' divides BB' in the ratio of BC to B'C'.
But since the figures are similar,
BC: B'C=AB : A'B';
that is, AA', BB', CC' meet in the same point S. In like manner it may be proved that DD' meets CC' in the point S.
:: AA', BB', CC', DD' are concurrent, and each of these lines is divided at S, externally or internally, in the ratio of a pair of homologous sides of the two figures.
Q.E.D. Cor. If any line is drawn through S meeting any pair of homologous sides in K and K', the ratio SK : SK' is constant, and equal to the ratio of any pair of homologous sides.
Note. It will be seen that the lines joining corresponding points are divided externally or internally at S according as the corresponding sides are drawn in the same or in opposite directions. In either case the point of concurrence S is called a centre of similarity of the two figures.
2. A common tangent STT' to two circles whose centres are C, C', meets the line of centres in S. If through S any straight line is drawn meeting these two circles in P, Q, and P', Q', respectively, then the radii CP, CQ shall be respectively parallel to C'P', C'Q'. Also the rectangles SQ . SP', SP. 'SQ' shall each be equal to the rectangle ST. ST'.
Join CT, CP, CQ and C'T', C'P', C'Q'. Then since each of the 48 CTS, C'T'S is a right angle,
VI. 7. . the L SCP=the SC'P';
CP is par to C'P'.
Similarly CQ is part to C'Q. Again, it easily follows that TP, TQ are parl to T'P', T'Q' respectively;
:: the As STP, ST'P' are similar. Now the rect. SP . SQ=the sq. on ST;
III. 36. .: SP : ST=ST : SQ,
VI. 16. and SP : ST=SP : ST';
ST : SQ=SP : ST'; :: the rect. ST. ST'=SQ . SP'. In the same way it may be proved that the rect. SP . SQ' = the rect. ST. ST'.
Q.E.D. CoR. 1. It has been proved that
SC : SC'=CP : C'P'; thus the external common tangents to the two circles meet at a point S which divides the line of centres externally in the ratio of the radii
. Similarly it may be shewn that the transverse common tangents meet at a point S' which divides the line of centres internally in the ratio of the radii.
COR. 2. CC' is divided harmonically at S and S'.
DEFINITION. The points S and S' which divide externally and internally the line of centres of two circles in the ratio of their radii are called the external and internal centres of similitude respectively.
EXAMPLES ON CENTRES OF SIMILITUDE.
1. Inscribe a square in a given triangle.
2. In a given triangle inscribe a triangle similar and similarly situated to a given triangle.
3. Inscribe a square in a given sector of circle, so that two angular points shall be on the arc of the sector and the other two on the bounding radii.
4. In the figure on page 298, if Di meets the inscribed circle in X, shew that A, X, D, are collinear. Also if Al, meets the base in Y'shew that I, is divided harmonically at Y and .
5. With the notation on page 302 shew that O and G are respectively the external and internal centres of similitude of the circumscribed and nine-points circle.
6. If a variable circle touches two fixed circles, the line joining their points of contact passes through a centre of similitude. Distinguish between the different cases.
7. Describe a circle which shall touch two given circles and pass through a given point.
8. Describe a circle which shall touch three given circles.
9. C1, C2, C3 are the centres of three given circles ; S1, S1, are the internal and external centres of similitude of the pair of circles whose centres are C2, C3, and S2, S., S'z, Sz, have similar meanings with regard to the other two pairs of circles : shew that
(i) S C1, S.,C2, S':C, are concurrent ;
(ii) the six points S, S., S3, S1, S2, S'3, lie three and three on four straight lines. [See Ex. 1 and 2, pp. 400, 401.]
1. If in any straight line drawn from the centre of a circle two points are taken such that the rectangle contained by their distances from the centre is equal to the square on the radius, each point is said to be the inverse of the other.
Thus in the figure given on the following page, if o is the centre of the circle, and if OP.OQ=(radius), then each of the points and Q is the inverse of the other.
It is clear that if one of these points is within the circle the other must be without it.
2. The polar of a given point with respect to a given circle is the straight line drawn through the inverse of the given point at right angles to the line which joins the given point to the centre: and with reference to the polar the given point is called the pole.
Thus in the adjoining figure, if OP. OQ=(radius)?, and if through
P and Q, LM and HK are drawn perp. to OP; then HK is the polar of the point P, and P is the pole of the st. line HK with respect to the given circle: also LM is the polar of the point Q, and Q the pole of LM.
It is clear that the polar of an external point must intersect the circle, and that the polar of an internal point must fall without it: also that the polar of a point on the circumference is the tangent at that point.
1. Now it has been proved (see Ex. 1, page 251] that if from an external point P two tangents PH, PK are drawn to a circle, of which O is the centre, then OP cuts the
Н. chord of contact HK at right angles at Q, so that
OP. OQ=(radius)? ; :: HK is the polar of P with respect to the circle.
Def. 2. Hence we conclude that
The polar of an external point with reference to a circle is the chord of contact of tangents drawn from the given point to the circle.
2. If A and P are any two points, and if the polar of A with respect to any circle passes through P, then the polar of P must pass through A.
Let BC be the polar of the point A
Join OP; and from A draw AQ perp. to OP. We shall shew that AQ is the polar of P.
B Now since BC is the polar of A, :: the L ABP is a rt. angle;
Def. 2, page 391. and the L AQP is a rt. angle : Constr. .. the four points A, B, P, Q are concyclic; :: OQ. OP=OA. OB III. 36.
=(radius)?, for CB is the polar of A: P and Q are inverse points with respect to the given circle.
And since AQ is perp. to OP
.. AQ is the polar of P.
Q.E.D. NotE. A similar proof applies to the case when the given point A is without the circle, and the polar BC cuts it.
The above Theorem is known as the Reciprocal Property of Pole and Polar.
3. To prove that the locus of the intersection of tangents drawn to a circle at the extremities of all chords which pass through a given point within the circle is the polar of that point.
Let A be the given point within the circle. Let HK be any chord passing through A; and let the tangents at H and K intersect at P.
I. To shew that P lies on the polar of A.
ПВ Sir is the chord of contact of tangents drawn from P, :. HK is the polar of P. Ex. 1, p. 391.