3.

The Arithmetic, Geometric and Harmonic means of two straight lines may be thus represented graphically.

In the adjoining figure, two tangents AH, AK are drawn from any external point A to the circle PHQK; HK is the chord of contact, and the st. line joining A to the centre O cuts the Oce at P and Q.

Then (i) AO is the Arithmetic mean between AP and AQ: for clearly AO=(AP+AQ).

H

A

P

BO

Q

K

(ii) AH is the Geometric mean between AP and AQ:
for AH2 AP. AQ.

(iii) AB is the Harmonic mean between AP and AQ:

for OA. OB=OP2.

.. AB is cut harmonically at P and Q.

III. 36.

Ex. 1, p. 233.
Ex. 1, p. 360.

That is, AB is the Harmonic mean between AP and AQ. And from the similar triangles OAH, HAB,

OA: AH .. AO. AB

. the Geometric mean between two straight lines is the mean proportional between their Arithmetic and Harmonic means.

4. Given the base of a triangle and the ratio of the other sides, to find the locus of the vertex.

Let BC be the given base, and let BAC be any triangle standing upon it, such that BA AC-the given ratio:

it is required to find the locus of A.

Bisect the BAC internally and

externally by AP, AQ.

A

Then BC is divided internally at P, and externally at Q,
so that BP: PC=BQ: QC the given ratio;

.. P and Q are fixed points.

And since AP, AQ are the internal and external bisectors of the

< BAC,

..the PAQ is a rt. angle;

.. the locus of A is a circle described on PQ as diameter.

EXERCISE. Given three points B, P, C in a straight line: find the locus of points at which BP and PC subtend equal angles.

DEFINITIONS.

1. A series of points in a straight line is called a range. If the range consists of four points, of which one pair are harmonic conjugates with respect to the other pair, it is said to be a harmonic range.

2. A series of straight lines drawn through a point is called a pencil.

The point of concurrence is called the vertex of the pencil, and each of the straight lines is called a ray.

A pencil of four rays drawn from any point to a harmonic range is said to be a harmonic pencil.

3. A straight line drawn to cut a system of lines is called a transversal.

4. A system of four straight lines, no three of which are concurrent, is called a complete quadrilateral.

These straight lines will intersect two and two in six points, called the vertices of the quadrilateral; the three straight lines which join opposite vertices are diagonals.

THEOREMS ON HARMONIC SECTION.

1. If a transversal is drawn parallel to one ray of a harmonic pencil, the other three rays intercept equal parts upon it: and conversely.

2. Any transversal is cut harmonically by the rays of a harmonic pencil.

3. In a harmonic pencil, if one ray bisect the angle between the other pair of rays, it is perpendicular to its conjugate ray. Conversely if one pair of rays form a right angle, then they bisect internally and externally the angle between the other pair.

4. If A, P, B, Q and a, p, b, q are harmonic ranges, one on cach of two given straight lines, and if Aa, Pp, Bb, the straight lines which join three pairs of corresponding points, meet at S; then will Qq also pass through S.

5. If two straight lines intersect at A, and if A, P, B, Q and A, p, b, q are two harmonic ranges one on each straight line (the points corresponding as indicated by the letters), then Pp, Bb, Qq will be concurrent: also Pq, Bb, Qp will be concurrent.

6. Use Theorem 5 to prove that in a complete quadrilateral in which the three diagonals are drawn, the straight line joining any pair of opposite vertices is cut harmonically by the other two diagonals.

II.

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 par1 to A'B';

S

.. the ▲ SAB, SA'B' are equiangular;

.. SA: SA'AB : A'B';

Hyp.

VI. 4.

.. 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';

.. AA' and CC' divide BB' in the same ratio; 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 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', then the radii CP, CQ shall be respectively parallel to C'P', C'Q'. respectively, Also the rectangles SQ. SP', SP. SQ' shall each be equal to the rectangle ST. ST'.

P' Q

Join CT, CP, CQ and C'T', C'P', C'Q'.
Then since each of the CTS, C'T'S is a right angle,
.. CT is par1 to C'T';

.. the As SCT, SC'T' are equiangular;
.. SC: SC'=CT : C'T'

=CP :CP′;

.. the SCP, SC'P' are similar;

.. the

SCP the SC'P';

.. CP is par1 to C'P'.

Similarly CQ is par1 to C'Q'.

III. 18.

VI. 7.

Again, it easily follows that TP, TQ are par to T'P', T'Q' respectively;

8

.. the STP, ST'P' are similar. Now the rect. SP. SQ=the sq. on ST;

III. 37.

VI. 16.

.. SP: ST

ST : SQ,

and SP ST

SP' : ST';

.. ST SQ

SP' : ST';

SQ. SP'.

that

.. the rect. ST.ST'

In the same way it may be proved

the rect. SP. SQ'

COR. 1. It has been proved that

the rect. ST. ST'.

SC: SC' CP : C'P';

Q. E. D.

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.

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 278, 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 Ấ.

5.

With the notation on page 282 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; S'1, S1, are the internal and external centres of similitude of the pair of circles whose centres are C2, C3, and S'2, S2, S3, S3, have similar meanings with regard to the other two pairs of circles: shew that

(i) S'C1, S2C2, S'3C3 are concurrent ;

(ii) the six points S1, S2, S3, S'1, S2, S3, lie three and three on four straight lines. [See Ex. 1 and 2, pp. 375, 376.]

III. ON POLE AND POLAR.

DEFINITIONS.

(i) 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 below, if O is the centre of the circle, and if OP. OQ=(radius)2, then each of the points P 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.