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2o. The middle points of the diagonals AX, BY, CZ of a complete quadrilateral are collinear.

3°. The line of collinearity of the orthocentres is at right angles to the line in 2°, called the Diagonal Line of the Quadrilateral.

4°. The circles on the three diagonals as diameters are coaxal.

5°. The polar circles of the four triangles belong to the conjugate coaxal system.

EXAMPLES.

1. A, B, C, D are the vertices of a convex quadrilateral taken in order; Ae, Be, Ce, De and A, B, C, D; the external and internal bisectors of the angles; prove that

ao. The sixteen centres of the circles touching the sides of the four triangles formed by taking the sides of the quadrilateral in triads, lie in fours on these bisectors.

B. The following groups of quadrilaterals are cyclic

Ae Bi Ci Del(a)

Ai Be Ce Di

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Ai Ce De
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A, B, C, D. } (c).

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7. Groups (a) and (c) are coaxal, and groups (b) and (d) conjugately coaxal.

[These properties are proved by employing Euc. III. 32 to show that any circle of either group is cut orthogonally by any circle of the other group. Russell.]

2*. A, B, C, D are four points on a circle. Omitting each point in turn we have four triangles; prove that the sixteen centres of the circles touching the sides of these triangles lie in fours on four parallel lines, and also in fours on four lines each perpendicular

* Educational Times, Reprint Vol. LI. p. 65.

to the former set; and that the two sets of lines are parallel to the bisectors of the angle between AC and BD. (M'Cay.)

3. ABC is a triangle, AA' a diameter of the circum-circle and H the orthocentre; show that A' and H are equidistant from the base BC; and hence deduce the theorem "the Simson line of any point is equidistant from the point and orthocentre of the triangle.”

SECTION II.

ADDITIONAL CRITERIA OF COAXAL CIRCLES.

92. I. Relation connecting the distances between the centres and the radii of three circles of a coaxal system. Let the circles be denoted by A, r1; B, r1⁄2; C, 13. Then for any point P on the radical axis, we have

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BC. AP2+CA.BP2+AB.CP2 — — BC. CA. AB; hence if t be the length of the tangent from P to the circles, since AP2=r12+t2, etc., by substituting in this equation and reducing,

BC.r2+CA.r22+AB. r2=-BC.CA. AB,.....(1) a result from which the radius r of any circle of the system may be found when the position of its centre is known; and conversely.

COR. 1. If r=0, C is a limiting point (Art. 87), by letting AC=x in (1) we obtain a quadratic in x, the last term of which is r2. Hence the product of the distances of the limiting points from the centre of any circle of the system the square of its radius. Cf. Art. 87.

COR. 2. If r2=r=0, the criterion reduces to
AB. AC=r2.

EXAMPLES.

1. If t1, t2, ta denote the tangents from any point P to three circles of a coaxal system; to prove that

BC.t2+ CA. t2+AB. t2=0.

[For BC. AP2+ CA.BP2+AB. CP2-BC. CA. AB,.....(1) and BC.r2+CA. r22+AB. rg2 = - BC. CA.AB.. .....(2) Subtracting (2) from (1); therefore, etc.]

2. Deduce as a particular case of Ex. 1 the theorem :-The locus of a point, the tangents from which to two given circles are in a constant ratio, is a coaxal circle.

[Let tз=0.]

3. Explain the formula of Ex. 1 when t2=ts=0.

4. Find the locus of a point P if the product of the tangents from it to two circles bears a constant ratio to the square of the tangent to any circle coaxal with them (ktit2=t32).

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[In Ex. 1, substituting the given condition, the equation reduces to the form (tı — mt2)(tı — nt2)=0; hence P describes two coaxal circles, since the ratio of the tangents t1 and t=m, or n.]

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5. If the common tangent ZZ' to two circles meet a coaxal circle in the points A and B; to prove that MZ and MZ' are the bisectors of the angles subtended by the chord AB at either limiting point.

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[For AZ, AM and BZ, BM being pairs of tangents drawn from two points A and B on the same circle to two circles of the system, it follows that AZ AM=BZ BM, by alternation AM|BM=AZ/BZ, and for a similar reason = AZ'|BZ'; therefore, etc.]

6. To describe two circles of a coaxal system touching a given line.

[In Ex. 5 divide the line AB internally and externally in Z and Z' in the given ratio AM/BM; therefore Z and Z' are the required points of contact. It will be noticed that the circles lie one on each side of the radical axis.]

7. A triangle ABC is inscribed in a circle of a coaxal system; prove that the points of contact X, X', Y, Y', Z, Z' of the three pairs of circles of the system which touch the sides BC, CA, and AB respectively,

a. Lie three and three on four lines,

B. Connect with the opposite vertices by six lines, passing three and three through four points.

[Apply the relations in Ex. 5 to the three sides; therefore, etc. Arts. 62 and 63.]

8. Apply the criterion of the Article to show that the nine-points-, circum- and polar circles are coaxal.

9. If points B and D are taken on any two circles whose centres are O and O' and joined to the limiting point M such that BMD is a right angle, the locus of the intersection of tangents at B and D to the circles is a coaxal circle.

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[Let the line BD meet the circles again in A and C ; then

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by (1), a constant quantity; therefore, etc. (cf. Art. 89, Ex. 8).]

10. A quadrilateral PQRS is inscribed to one circle and escribed to another at the points A, B, C, D; prove that its position is indeterminate, and the diagonals PR and QS, BC and AD of the two cyclic quadrilaterals intersect (the latter at right angles) at the limiting point M.

[By Art. 89, Ex. 6. See also Art. 88, Ex. 1, and Art. 67, Cor. 6.]

11. Construct a quadrilateral in a given circle symmetrical with respect to a given diameter and circumscribed to a circle having its centre at a fixed point on the diameter.

[Find the radius of the second circle by Art. 89, Ex. 6.]

93. II. A variable circle cuts three others of a coaxal system at angles a, B, y, to prove the relation

BC.rcos a+CA. r,cos ẞ+AB. r2cos y=0.

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Let P, p be the variable circle meeting the given ones at the points R, S, T respectively; join PR, PS, PT, and produce the lines to meet the circles again in R', S', T.

By Art. 92, Ex. 1, BC. t2 + CA. t2 + AB.t2=0, but t12=PR. PR' = p(p+RR')=p(p+2r,cos a), with similar values for t2 and t3. Substituting these values in the equation and reducing, we obtain the required result.

COR. 1. If two of the circles are cut orthogonally, every circle of the system is eut orthogonally. For if a=B=90°, two terms of the equation vanish, therefore AB. r,cos y=0 or y=90°.

COR. 2. If the variable circle touch two of the given

it cuts the circle C, r, coaxal with them at an angle

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