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[For we have PA-PA', PB PB', QA-QA' and QB-QB'; hence PA/QA-PB/QB etc.; therefore, etc.

The second part follows, since MN is divided harmonically by P and Q. Art. 70.]

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13a. To what does the theorem reduce when AA' and BB' coincide?

14. For any two pairs of inverse points P, Q and P', Q' prove that

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[PPQQ is a cyclic quadrilateral (Ex. 4); hence the triangles OPP' and OQQ' are similar; so also are OPQ' and OP'Q; therefore, etc. (Euc. VI. 4). Otherwise if p and q denote the perpendiculars

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15. If P, Q, R be any three collinear points on the diagonal triangle of a quadrilateral; their harmonic conjugates P'Q'R' with respect to the diagonals XX', YY', ZZ' are also collinear. [For XX' is divided harmonically in B and C (Art. 68) and P and P; hence, by Ex. 14,

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BP CQ AR BP' CQ' AR' BL CM AN CP AQ BR' ̃ ̄CP'' AQ'` BR' CL AM BN but P, Q, R are in a line; † therefore, etc.]

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16. To what does Ex. 15 reduce when the line PQR is at infinity?

17. The angles subtended by the diagonals of a complete quadrilateral at any point O have a common angle of harmonic section, real or imaginary.

[O is the point of intersection of the lines PQR and P'Q'R' in Ex. 16; therefore, etc.]

18. The circles on the diagonals of a complete quadrilateral pass through two points, real or imaginary.

[In Ex. 17, if two of the angles XOX', YOY' are right; ZOZ' must also be right,‡ since it is divided harmonically by PQR and P'Q'R'.]

19. Any transversal to the pencil in Ex. 17 is cut in six points which, taken in pairs, have a common segment of harmonic section. 20. To what does Ex. 17 reduce when O is at infinity?

21. If the sides of a triangle ABC are divided harmonically in XX', YY', ZZ' ; if X, Y, Z are collinear, the middle points L, M, N of these segments are collinear.

22. If perpendiculars be let fall on the sides of a triangle from a pair of inverse points O and O' and their feet joined; the triangles PQR and P'Q'R' thus formed are similar and their areas are as the distances of O and O' from the circum-centre.

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* Hence also the middle points L, M, N of the diagonals of a complete quadrilateral are collinear.

+ PQR and P'Q'R' are termed Conjugate Lines of the quadrilateral. Generally, For a number of angles at a common vertex having a common angle of harmonic section if any two are right, all the others are also right.

23. Through a point P in the diameter of a semi-circle draw a chord AB such that the area of the quadrilateral ABA'B', where A'B' is the projection of AB on the diameter, may be a maximum.

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[Let Q' be the inverse of P with respect to the circle; draw QQ' at right angles to A'B'. Project M the middle point of AB on A'B' and let X be the intersection of MM' with the semi-circle on Q'O. Then area S of quadrilateral ABA'B' = A'B'. MM', hence

S2=4MM'2. A'M'2=40M'. PM'. M'P. M'Q', by Art 70,
=4PM". OM'. M'Q=4PM'2. M'X2;

or S is equal to the area of the maximum rectangle that can be inscribed in a given circle, one of whose sides is parallel to a given line. Art. 14, Ex. 2.]

24. Six perpendiculars are drawn from the inverse of the intersection of the diagonals of a cyclic quadrilateral to the sides and diagonals. Show

1°. The feet of those to the sides are collinear.

2°. The line of collinearity bisects at right angles the line joining the feet of perpendiculars on the diagonals.

[By method of Ex. 22.]

25. If XX'; YY'; ZZ' denote the feet of the bisectors of the angles of a triangle ABC, show that the pedal triangles of two points and O' inverse to any of the circles on these segments as diameters, with respect to ABC, are inversely similar. (Neuberg.)

[Let O and O' be inverse with respect to ZZ'C,* PQR and Q'PR' their pedal triangles respectively. M the middle point of ZZ'.

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also the angles R and R' are equal; therefore etc.

NOTE.-If O is on the circle ZZ'C the pedal triangle is isosceles, similarly if it is the point of intersection of the circles ZZ'C and YY'B it is isosceles in a double aspect, i.e. equilateral.

Hence we may infer that the circles AXX', BYY', and CZZ' pass through two points O and O' which are inverse (Ex. 22) with respect to the circum-circle of ABC and whose pedal triangles with respect to ABC are equilateral.]

* Le cercle d'Apollonius du triangle ABC par rapport à AB. V. Educ. Times, Dec., 1890.

CHAPTER VII.

POLES AND POLARS WITH RESPECT TO A CIRCLE.

SECTION I.

CONJUGATE POINTS, POLAR CIRCLE.

73. Def. The perpendicular to the line joining a pair of inverse points passing through either is the Polar of the other with respect to the circle. In the figure of Art. 74 C and Z are inverse points; and C and the line AB are termed Pole and Polar with respect to the circle. Any point A or B on the polar is the Conjugate of C, hence the polar of a point is the locus of its conjugates.

Again, since the circle on BC as diameter passes through Z and therefore cuts the given one orthogonally :1°. The circle described on the line joining two conjugate points cuts the given circle orthogonally. 2°. The distance between two conjugate points is equal to twice the length of the tangent to the circle from the middle point of the line connecting them.

74. Theorem. For any two conjugate points B and C, to prove that each lies on the polar of the other with respect to the circle.

Suppose the polar of C to be AB, we require to prove that the polar B passes through C. Join AO, draw a

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