SECTION vii-HARMONIC RANGES.

Def. If the segment AB of a line is divided internally in X and externally in Y, in the same ratio, so that

then the four points A, X, B, Y are termed a harmonic range; and the pair of points X and Y are termed harmonic conjugates of each other with respect to A and B.

it follows that A and B are harmonic conjugates of each other with respect to X and Y.

Note (2)-Since AX. BY

=

BX. AY, the cross-ratio (AXBY) is unity, and .. constant. Hence all Theorems deduced from the constancy of crossratios are true for harmonic ranges.

Def. Three magnitudes are said to be in harmonic proportion when

And the 2nd is termed the harmonic mean between the 1st and 3rd.

The words harmonic range and harmonic mean will be respectively abbreviated into H. R. and H. M.

THEOREM (1)—If four points form an H. R. the distance of either extreme point from its own conjugate is an H. M. between its distances from the other

Cor.

Since AY (AB - AX)

=

AX (AY - AB);

..

AB (AX + AY) = 2 AX. AY.

Simrly. XY (AY + BY) = 2 AY. BY.

And, conversely, if either of these relations holds, the points A, X, B, Y form an H. R.

THEOREM (2)-If X, Y are harmonic conjugates with respect to A, B ; and M is the mid point of AB; then MA3 MB2; and conversely.

=

MX. MY =

whence

=

or 2 MA : 2 MX = 2 MY: 2 MA,

MA2 = MX. MY.

The converse follows by simply retracing the preceding steps.

Cor. X and Y move in opposite directions.

THEOREM (3)-If the points of section of a pencil of four rays by any one transversal form an H. R., then the points of section of the pencil by every transversal will form an H. R.

Now let abcd be any other transversal to the pencil; and pcq || to PCQ;

where similar letters are on same ray.

Def. A pencil through two pairs of harmonic conjugates is called a harmonic pencil.

Cor. (1). The intercepts made by three adjacent rays of a harmonic pencil, on a parallel to the fourth ray, are equal.

Cor. (2). O (ABCD) is harmonic if a transversal parallel to a ray is cut into equal segments by the remaining three.

THEOREM (4)—The arms of an angle, and its internal and external bisectors form a harmonic pencil; and conversely, if in a harmonic pencil the angle between a pair of rays is right, then these rays are the internal and external bisectors of the angle between the other two.

The first part follows at once from vi. 3, in connection with the def. of an H. R.

i.e. OB is internal bisector of AOC.

Also OD, being to OB, is external bisector of AỐC.

B b

THEOREM (5)—In a complete quadrilateral, if the intersections of the three diagonals are joined, and the joins produced; then all ranges and pencils are harmonic.

.. X (AQBY) and O (AQBY) are harmonic.

And.. so also are X (AOCZ), Y (AOCZ), Y (ASDX),, &c.

Note-The preceding Theorem suggests a mode of drawing the fourth ray of a harmonic pencil, when three consecutive rays are given. For let XA, XQ, XB be three given rays; take O any pt. in XQ, and let BO, AO meet XA, XB in D, C ; let DC meet AB in Y; then XY is the fourth harmonic ray.

Cor. The Theorem is also true for the more extended definition of a complete quadrilateral given in the Note on p. 299.

EXERCISES ON HARMONIC RANGES.

1. The join of the points of contact of two sides of a triangle with its in-circle, meets the third side at the harmonic conjugate of the third point of contact, with respect to the two corners in that side.

NOTE-Use Menelaus' Theorem.

2. A similar Theorem to the last holds for each of the ex-circles.

3. Through any point in an altitude of a triangle lines are drawn from the ends of the base; if the points in which these lines meet the opposite sides are joined to the foot of the altitude, the joins make equal angles with the altitude. NOTE-iii. Addenda (19) is a particular case of this.

4. From a fixed point two variable transversals are drawn to two fixed intersecting lines; if the points of section are joined transversely, find the Locus of the intersection of the joins.

5. If a line is drawn across a pair of orthogonal circles, it is harmonically divided by the circumferences if it goes through the centre of either.

6. Conversely to the last Exercise-If a circle is drawn through a pair of harmonic conjugates with respect to the ends of a diameter of another circle, then the circles cut orthogonally.

7. If a line touches two circles, then any circle co-axal with them cuts it in points which are harmonic conjugates with respect to the points of contact.

8. Any point on the circumference of a circle is joined to the ends of a chord ; show that the joins (produced if necessary) cut the diameter perpendicular to the chord in points which are harmonic conjugates with respect to the ends of that diameter.

9. If X, Y are a pair of harmonic conjugates with respect to the ends A, B of a diameter of a circle, and P is any point in the perpendicular to AY at Y; then PX is cut harmonically by the circle.

10. If a transversal is drawn to a triangle, so as to bisect one of its sides, then the parallel to the bisected side, through the opposite corner, meets the transversal in a point which forms a harmonic range with the three points in which the transversal cuts the sides.

II. Two circles cut in A, B; if XX' is any diameter of one, and YY' any diameter of the other; and if XY, X'Y' meet in Z, and XY", X'Y in Z'; then the circle on ZZ' as diameter goes through A, B.

NOTE-Use Theorems (5), (4) and Menelaus'.