hence M' is the mean centre of X' and C', that is of A', B' and C', for the same multiples that M is of A, B and C.

But each of the ratios in (1) is equal to M'O/M'R; therefore M' is the mean centre of O and R, that is of P, Q and R for the same set of multiples.

NOTE.-The construction for the displaced mean centre may in the same manner be extended to the quadrilateral and generally to a polygon of any number of sides.

Hence for two systems of points A, B, C, ... and A', B', C', ... and their mean centres M and M' for the same set of multiples a, b, c... if we draw through M parallels MP, MQ, MR, ... equal to AA', BB', CC',... respectively, the mean centre of the third system P, Q, R,... for the same multiples coincides with M'.

9. If through any point M are drawn MP, MQ and MR parallel and proportional to the sides of a triangle ABC, the mean centre of P, Q and R for multiples each equal to unity coincides with M.

[By Ex. 8, or thus :-Complete the parallelograms PMQR' and draw MR.

Since

PM PM α

=

PR QM b

and the angles at P and C equal, the tri

angles PMR' and ABC are similar, hence MR MR'=2MO, and O

is the mean centre of P and Q, and therefore M is the mean centre

of P, Q, R.]

10. Prove the similar property for the quadrilateral; and generally :

If through any point M lines are drawn parallel and proportional to the sides of a polygon; the mean centre of their extremities for multiples each=1 coincides with M.

...

11. If a system of points A, B, C,... be displaced to A', B', C', ... such that AA', BB', CC',. are parallel and proportional to the sides of a polygon, the mean centre of the system remains a fixed point. [By aid of Exs. 8 and 10.]

12. Weill's Theorem.-A variable polygon is inscribed to one circle and escribed to another; to prove that the mean centre of the points of contact of its sides with the latter circle is a fixed point.

[Let ABC... denote the polygon, A'B'C'... a consecutive position, T and T' the points of contact of AB and A'B' with the circle of radius r; the small angle between AB and A'B', and X their intersection.

The triangles AA'X and BB'X are similar, hence BB'|AA'=BX/A'X

Also, since AB and A'B' are indefinitely near to one another, X is indefinitely near to the point of contact T, and BX and BY are therefore equal because they are tangents from the same point to a circle.

Thus as the polygon ABC... varies, its points of contact are displaced for each consecutive position in the direction of its sides, and proportional to them; therefore the mean centre is a fixed point.

NOTE. If the side BC is a variable tangent to a third circle of radius r, the result of dividing (2) by (1) is

therefore if the three circles are so related that BX BY is a constant

13. The mean centres of the vertices of any polygon and of similar triangles similarly described on its sides coincide (M‘Cay). [Let the vertices of the triangles on the sides AB, BC, CD be A', B, C'... respectively.

...

Since AA': BB' ; CC' ... = AB; BC: CD ... and are inclined to the sides of the polygon at the same angle; we may regard the vertices of the given polygon displaced to A'B'C'... distances proportional and parallel to its sides turned through that angle (cf. Ex. 6).]*

* The proofs of Examples 11-13 were communicated to the Author by Mr. Charles M'Vicker.

G

14. Through the centre O of a regular polygon any line is drawn meeting the sides in A', B', C',... to prove that Σ

1

OA'

=

=0.

[Let M be the middle point of one side, then MA'O is a rightangled triangle, and if a perpendicular MM' be let fall on the hypotenuse we have

OA'. OM' = r2 or Σ

1

= OA'

ZOM'=0. Art. 50. See Art. 3, Ex. 9.]

54. Theorem. For any system of points A, B, C,... their mean centre O, and any line L; to prove that Za. AL2 Za. AL2+2(a)OL2,

=

where L' is the line through O parallel to L.

For AL-AL+OL; .·. AL2=AL22+OL2+2AL'.OL; BL=BL+OL; .. BL2 = BL2+0L2+ 2BL'. OL;

Multiplying these equations by a, b, c,... respectively and adding results,

Za. AL2=Za. AL22+Σ(a)OL2+20LΣ(α. AL'),

but Ea. AL=0 (Art. 50); therefore, etc. COR. 1. When the multiples are equal ZAL2=ZAL2+nOL2,

also since ΣAL=n. OL; OL is the arithmetical mean of the several lines AL, BL, CL ..., and AL', BĽ... the several differences between each and their mean.

Hence, the sum of the squares of n quantities=n times the square of their mean value+the sum of squares of the n differences; or if the quantities are the segments of a line this property may be stated: the sum of the squares of the unequal parts = the sum of the squares of the equal parts+the sum of the squares of the n differences. This property is obviously an extension of Euc. II. 9, 10.

COR. 2. For any two parallel lines L and M,

Za. AL-Za. AM2 = X(a)(OL2 — OM2).

55. Theorem. For any point P to prove that Za. AP2 Za. A02+2(a)OP2.

=

Project the system of points on the line OP and denote their projections by A', B, C,.... Then (Euc. III. 12-13),

Similarly

BP2-B02+0P2+20P. OB' etc.

Multiplying these equations by a, b, c... and adding the results,

Σα . ΑΡ2 = Σα .Α0' + Σ(α)OP + 2ΟΡΣα . ΟΑ',

but O is the mean centre of the system A', B, C... (Art. 53, Ex. 4); therefore Za. OA'=0.

COR. 1. If the n multiples are equal

ZAP2=ZA02+n. OP2.

COR. 2. For a regular cyclic polygon the sum of the squares of the distances of any point on the circle from the n vertices is constant and = 2n R2.

COR. 3. If Za. AP2 is constant, the locus of P is a circle concentric with O the square of whose radius is Za. A P2-Za. A O2 equal to

Σ(α)

COR. 4. Za. AP2 is a minimum when P coincides with O. See Art. 16, Ex. 3.