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13.

To find the condition that three straight lines may intersect in a point.

Let the equations of the straight lines be

1⁄4 ̧a+m1ß +n ̧y = 0,

12a+m‚ß+n,y= 0,

l ̧α +m ̧ß+n ̧y = 0.

If these three straight lines intersect in a point, the above three equations must be satisfied by the same values of a, ß, y.ˆ This gives, eliminating a, B, y by cross-multipli

cation,

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The identity of form between the conditions that three straight lines should intersect in a point, and that three points should lie in a straight line, is worthy of notice. Its full geometrical meaning will be seen hereafter.

We shall sometimes, in future investigations, speak of the straight line represented by the equation la + mẞ+ny = 0, as the straight line (l, m, n). Adopting this phraseology, it will be seen that the condition that the three points (1,, m,, n,) (1,, m,, n) (13, m3? n) should lie in the same straight line, is the same as the condition that the three straight lines (1, m ̧, n) (1,, m, n,) (1, m, n) should intersect in a point.

14. To find the condition that two straight lines may be parallel to one another.

Let the equations of the two straight lines be

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Let (f, g, h) (a, B, y) be the co-ordinates of any two points in (1),

(f', g', h') (a', B', y) be the co-ordinates of any two points in (2).

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Also, recurring to the investigation of Art. (8), fig. 7,

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These two equations are equivalent to one only, since they may be written in the form

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where it will be seen that the equality of any two members implies that the third is equal to either of them.

Multiplying the numerators and denominators of the several members of (3) by l', m', n' and adding, we obtain the condition under the form

(mn' — m'n) a + (nl' — n'l) b + (lm' — l'm) c =
c = 0......(4).

This is the necessary condition of parallelism, and is generally the most convenient form which can be employed. It is equivalent to

(mn' - m'n) sin A+ (nl' — n'l) sin B+ (lm' — l'm) sin C=0,

a form which we shall occasionally use.

It will be observed that this condition is the same in form as that which results from the elimination of a, B, y between the equations

la + mß + ny = 0,

l'a+m'B+ n'y = 0,

aa+bB+cy = 0.

The last of these is, as we know, an equation which cannot be satisfied by any values of a, B, y, since, as we have already proved (Art. 2), aa +bB+cy=2A. Hence the equation (4) may be looked upon as an expression of the fact that two equations

la + mB + ny = 0,

l'a+m'B+ n'y = 0,

cannot be simultaneously satisfied by any values of a, ß, y, or, in other words, that the two straight lines represented by them do not intersect, which is known to be a necessary condition of their parallelism, and also a sufficient condition, since the two straight lines are in the same plane.

Although, however, no values of a, ß, y exist which will satisfy the equation aa+bB+cy = 0, yet we can always satisfy the equation la + mẞ+ny = 0, where the ratios : m : n approach as nearly as we please to the ratios a: b: c.

By referring to the investigation of Art. (7) it will be seen that, q, r, denoting the distances from A, of the points in which the straight line (1, m, n) cuts AC, AB respectively,

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It hence appears, that by making the ratios 7: m : n sufficiently nearly equal to the ratios ab: c, the values of q and may be made as great as we please, in other words, that the straight line (7, m, n) may be removed as far as we please from the triangle of reference. The limiting position,

therefore, to which the straight line (1, m, n) continually approaches, and with which it ultimately coincides, when the ratios 1: m:n continually approach to, and ultimately coincide with, the ratios a: b: c, is a straight line altogether at an infinite distance.

This is often expressed by saying that the equation

aa+bB+cy = 0,

or the equivalent equation

a sin A+B sin B+ y sin C=0,

represents the straight line at infinity.

This phraseology is very convenient, and free from objection, if the conventions on which it is adopted be clearly understood. It is, however, desirable that attention should be called to the fact, that the equation

aa+bB+cy=0

is, in itself, impossible,-in fact, a contradiction in terms,and can only be admitted as a limiting form to which possible equations may continually tend.

15. To find the equation of a straight line, drawn through a given point, parallel to a given straight line.

Let (l, m, n) be the given straight line, (f, g, h) the given point, then the equation of the required straight line will be

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For this straight line passes through the point (f, g, h), and does not intersect the straight line (l, m, n); since, if it did, we should have aa+bB+cy = 0.

Since af+bg+ch = 2A, this equation may also be written

If+mg + nh

la + mB + ny=

2A

(aa+bB+cy).

COR. The general equation of a straight line parallel to (l, m, n) is

la+mB+ny = k (aa+bB+cy),

where k is an arbitrary constant.

16. To find the inclinations of a straight line, drawn through one of the angular points of the triangle of reference, to the sides which intersect in that point.

Let the equation of the straight line AP be

μβ=γ,
Fig. 8.

and let be its inclination to AD, the internal bisector of the angle 4.

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