L

COR. 8. If the point P in Cor. 1 is such that two pairs of opposite connectors PX, PX'; PY, PY' are at right angles, the tangents from P to the circle are likewise at right angles. But the circle reciprocates from P as origin into an equilateral hyperbola; therefore if an equilateral hyperbola be circumscribed to a triangle, it passes through the orthocentre.

More generally, if an equilateral hyperbola be described about a quadrilateral, it passes through the orthocentre of the four triangles formed by taking the vertices in triads.

The property of Art. 68, Ex. 8, will now appear obvious. It follows also that the locus of the centres of equilateral hyperbolas described about a triangle is its nine-pointscircle.

COR. 9. If the sides of the quadrilateral be numbered 1, 2, 3, 4, and the perpendiculars from W and W' on them be denoted by P1, P2 P3 P4; 91, 92, 93, 94, since

[WW'XX']=[WW'YY']=[WW'ZZ'],

hence P2P3/P1P4 is of constant value for all points on the conic, or the locus of a point such that the products of the perpendiculars from it to the three pairs of opposite sides of a quadrilateral have constant ratios is a conic passing through its vertices; and by reciprocation we derive the correlative theorem:-If a quadrilateral is circumscribed to a conic, the rectangles under the distances of the pairs

of opposite vertices from a variable tangent have are to each other in constant ratios.*

COR. 10. If either asymptote of a hyperbola be taken as a transversal to an inscribed quadrilateral, the double points of the involution are both at infinity, and the segments XX', YY', ZZ have a common middle point; therefore the lines joining a variable point on a hyperbola to a pair of fixed points on it intercept segments of constant length on each of the asymptotes.

This property is thus stated in Townsend's Modern Geometry, Art. 340:

"For every two homographic pencils of rays through different vertices there exist two lines, real or imaginary, on each of which the several pairs of corresponding rays intercept equal segments.'

EXAMPLES.

1. A pencil whose rays are parallel to the three pairs of opposite connectors of a quadrilateral determines a system in involution. [Since the line at infinity is a transversal cut in involution by the sides of the quadrilateral; therefore, etc.]

2. The three pairs of parallels drawn through the vertices and the extremities of the third diagonal of a quadrilateral cut any transversal in a system of points in involution.

3. If the fourth vertex D of the quadrilateral ABCD is the orthocentre of ABC, prove the following particular case of the general theorem of Art. 148:-For any pencil of rays in involution, if two pairs of conjugates are at right angles, then all pairs of conjugates are at right angles.

4. Hence deduce "The circles on the diagonals of a complete quadrilateral are coaxal.”

5. Any line or circle intersects a coaxal system at points in involution.

6. The parallels through any point to the sides of a triangle and the lines connecting that point to the vertices form an involution.

7. Every two circles and their two centres of perspective subtend at any point a pencil in involution.

8. For every two self-reciprocal triangles with respect to the same circle any two vertices connect equianharmonically with the remaining four.

CHAPTER XIV.

DOUBLE POINTS.

149. The solutions of a large number of problems of every variety in Geometry are frequently made to depend on the finding of the double points of two homographic systems. On account of the great importance of these points various constructions have been given for them. Thus in the last corollary they are easily found when we have obtained the points whose conjugates are at infinity on the axis by the equations

XA.X'A' XM.X'M=XN. X'N.

=

We give in the following article two additional constructions for homographic rows on an axis and append a sufficient number of examples, some of which have apparently no connexion with our present subject, to enable the student to form an idea of their extensive applications.

150. For any two systems of points on a circle (Art. 67, Ex. 6) the pairs of lines BC', B'C; CA', C'A; AB', A'B intersect respectively in points X, Y, Z, which are collinear; and the line of collinearity meets the circle in points M and N, real or imaginary, given by the equations [ABCM]=[A'B'C'M] and [ABCN]=[A'B'C'N].

But since the anharmonic ratios are unaltered by inversion, if the origin O be taken on the circle, the cyclic system inverts into points lying on a line and the double points of the former invert into the double points of the latter system.

Hence the following construction for the double points of two homographic systems ABC... and A'B'C'... on a line.

Take any arbitrary point 0 and describe the circles BOC', B'OC meeting again in X; COA', C'OA in Y; and AOB', A'OB in Z. Then O, X, Y, Z lie on a circle which meets the axis in the required points M and N, real or imaginary. (Chasles.)

Otherwise thus:-Since [BCAM]=[B'C'A'M], we have

BA |BM B'A' [B'M

CA CMCA' CM'

which gives on reduction the ratios MB. MC/MB'. MC, a known quantity.

But the numerator and denominator are respectively the squares of the tangents from M to the circles described on the segments BC' and B'C as diameters; therefore, etc., by Art. 88, Cor. 2.

It should be noticed that two homographic systems whose double points are imaginary may be generated by the revolution of a constant angle about either of two fixed vertices which are reflexions of one another with respect to the axis. For if AA', BB, and CC' subtend equal angles at a point P (Art. 72, Cor. 8), then