where is the projection of x on the parallel b; but by Ex. 1.

substituting for x' its value and for x2, a2+c2-2ac cos ac, the above result is obtained.]

4. Euler's Theorem.*-For three collinear points A, B, C and any fourth P to prove the relation

BC. AP+CA. BP2+AB. CP-BC. CA.AB.

[By Euc. II. 12, 13, AP2= A B2+BP2-2AB. BP cos B...........(1) and CP2-BC2+BP2 + 2BC. BP cos B...........(2) multiplying (1) by BC and (2) by AB and adding to eliminate cos B, the above follows on reduction.]

4A. Having given the base c of a triangle and la2+mb2 = const., find the locus of the vertex, 7 and m being given quantities.

5. If APC is a right angle the relation in Ex. 4 is equivalent to BC2. AP2+AB2. CP2- AC2. BP2.

[This follows from Ex. 4 or is obtained directly thus; let fall perpendiculars BX and BY on CP and AP, then

XY2=BP2 BX2+ BY2= BC2sin2C+A B2sin2A;

=

multiplying the equation BP2 BC sin C+AB'sin2A by AC2; therefore, etc.].

6. If the transversal to a harmonic pencil is parallel to one ray D, the intercept AC is bisected by B the conjugate of D.

* "Catalan's Théorèmes et Problèmes de Géométrie Élémentaire," 1879, p. 141.

7. If a line L turn around a fixed point P and meet two fixed lines OA and OB in A′ and B'; the locus of the harmonic conjugate

Q of P with respect to A'B' is a line passing through 0; and

Note. By Euc. VI. 2 if the variable PQ is bisected at Q' the locus of Q' is a parallel to OQ and

Hence for any three lines A, B, C we find in the same manner that

8. For any system of lines A, B, C, D ... the locus of such that

1

1 PB

1

1

PA+PE+PC+ ... = PQ (or 2p1=PQ)

is a right line. [See Exs. 6 and 7.]

PAPQ

9. For a regular cyclic polygon, if P coincides with the centre

[Through P draw the line parallel to one of the sides, etc.]

10. If parallels be drawn through any point O to the four lines in Ex. 4, the relation may be written

11. From the formula BC. AD+CA.BD+AB. CD=0, prove that if A, B, C be three collinear points and P any fourth point BC cot A+ CA cot B+AB cot C=0, the angles being all measured in the same aspect; and hence find the locus of the vertex, having given the base c and cot A+m cot B-const.

4. Limiting Cases. 0 and ∞.

Def. The Angle of intersection of two circles is that between the tangents drawn to them at either point of

intersection; it is therefore equal to the angle between the radii drawn to either common point.* (Euc. III. 19.)

If the circles touch Internally this angle is 0°, if Externally 180°. They are said to intersect Orthogonally when the angie is 90°.

The Angle made by a line and circle is that between the line and the tangent to the circle at its intersection.

EXAMPLES.

1. To find the angles between the circum- and ex-circles of a triangle ABC.

[Since 82 R2+2Rr, etc., we easily obtain 2 cos 01= with similar expressions for 02 and 03.]

2. To find the angle of intersection of the in- and circum-circles.

3. If two concentric circles cut orthogonally one is real and the other imaginary, and their radii are of the forms p, ip.

* If 01, T1; O2, 72, be the circles, & the distance 012, the angle of intersection, and t the direct common tangent, we have

Similarly

82 − (r1+r1⁄2)2 = − 4r1r1⁄2 cos2 † 0,

hence if t' be the transverse common tangent,

...(1)

(2)

.(3)

where √1=i; also if y denote the length of the common chord, of the circles (real or imaginary) since 2r1r2 sin @= yd, t. t' = i. y. d.

It is obvious that either the transverse common tangent to the circles or their angle of intersection is imaginary.

Let AX be a variable chord passing through a fixed point A at which a tangent is drawn. According as the

chord AX and angle TAX diminish in magnitude X approaches the tangent. When X is indefinitely near to A, AX is said to have reached its limiting position and may then be considered to coincide with the tangent.

Hence a tangent to a circle is in the direction of the infinitesimal chord at its point of contact, or is the chord joining two indefinitely near points.

Again, let the tangent T and its point of contact be fixed and the chord AX given in length. As the radius of the circle increases the curvature diminishes, and the point X obviously approaches the tangent. Hence X may be made to move as near as we please to the tangent by continually increasing the value of the radius of the circle.

In the limit, when the latter is indefinitely great, the distance of X from T is so very small that we may consider the point to lie on the line. Hence a finite portion of a circle of indefinitely great radius opens out into a right line, the remainder being, of course, at a distance infinitely great, i.e., at infinity.

5. Envelopes.-Let a variable line turn around a fixed point O and meet any fixed line.

According as its angle of inclination to the perpendicular OM increases, the segments OA, OB, OC continue to increase and the angles A, B, C to diminish. In the

limit it reaches a position at right angles to OM. Here the angle between it and the fixed line vanishes, and their point of section is at infinity. In this case the lines are parallel (Euc. I. 28); hence

Parallel lines may be regarded as having angles of inclination = 0° or lines intersecting at infinity. Thus a system of parallels is a pencil of rays whose vertex is at infinity.

6. Let A and X be any two points on a curve of which A is fixed and X variable, and TA and TX tangents. It appears as before that as X approaches A the chord AX and the base angles A and X of the triangle TAX gradually diminish and ultimately vanish.

But as the base angles diminish the vertex T approaches the base and a fortiori the element of curve AX. Hence in the limiting position, i.e., when the tangents are consecutive, their point of intersection is on the curve.

A curve touched by a variable line is called the Envelope of the line. Thus the envelope of a line which