Hence, the inclinations of the given straight line to AB, AC, are determined. 17. To find the condition that two given straight lines may be perpendicular to one another. Let (l, m, n), (l', m', n') be the two given straight lines. Through A draw two straight lines parallel to them. These will be represented by the equations And these straight lines must be at right angles to one another. If 0, ' be the respective inclinations of these straight lines to the internal bisector of the angle A, then, by the result of the last article, And, if these be at right angles to one another, + {(lc-na) (m'a- l'b) + (ma — lb) (l'c — n'a)} cos A = 0; .. ll' (b2 + c2 — 2bc cos A) + mm'a2 + nn'a2 — (mn' + m'n) a2 cos A − (nľ' + n'l) (ac — ab cos A) 2 which, since b2+c2 - 2bc cos A= a, c-b cos A = a cos B, b-c cos A =a cos C, reduces to ll'+mm'+nn'—(mn'+m'n)cosA—(nl'+n'l)cosB—(lm'+l'm)cos C=0, the required condition. 18. To find the perpendicular distance from a given point to a given straight line. Let (f, g, h) be the given point, (7, m, n) the given straight line. Then, if q and r be the distance from A, of the points where this straight line meets AC, AB, respectively, we have shewn (Art. 7) that Now, let a denote the distance from (f, g, h) to (1, ́m, n). Then (q2 + r2 — 2gr cos A)¿ a′ +qg+rh=27 (af+bg+ch), bc 1 + 2 r 4 2 cos A α l*b*c* H|+ qr = a (l cos C+ n cos A – m) α lbc ; (lc −na) (lcos B+mcos A−n) +(lb-ma) (l cos C + n cos A – m) m)} {12 (c cos B+b cos C) + m2a + n3a — 2mn a cos A nl (c + a cos B− b cos A) — lm (b − c cos A + a cos C)}, which, by reduction, is equal to (l2 + m2 + n3 − 2mn cos A - 2nl cos B – 2lm cos C). (12 + m2 + n2 − 2mn cos A – 2nl cos B-2lm cos the required expression. It will be observed, that the numerator of this expression vanishes if the point (f, g, h) lie upon the line (l, m, n), as manifestly ought to be the case. It will also be remarked, that the more nearly the ratios 1 m n approach to the ratios ab: c, the less does the denominator of the above fraction become, and the greater, therefore, the distance from the point to the line; which is in accordance with the remark made in Art. (14). EXAMPLES. 1. Find the equation of the straight line joining the middle points of two sides of the triangle of reference; and thence prove that it is parallel to the third side. 2. Find the equations of the straight lines, drawn through the several angular points of the triangle of reference, respectively at right angles to and thence prove that they intersect in a point. 3. If 0 be the angle between the two straight lines (1, m, n), (A, μ, v), prove that co = lλ+mμ+nv−(mv+nμ) cos A − (nλ+lv) cos B − (lμ+mλ) cos C (mv – nu) sin 4 + (n) −lv) sin B + (u – mà) sin C 4. On the sides of the triangle ABC, as bases, are constructed three triangles A'BC, AB'C, ABC", similar to each other, and so placed that the angle BA'C = B'AC = BAC', CB'A = C'BA = CBA', AC'BA'CB = ACB'. Prove that the straight lines AA', BB', CC' intersect in one point. 5. Prove that the straight line, joining the centre of the circle inscribed in the triangle ABC, with the middle point of the side BC, is parallel to the straight line joining A with the point of contact of the circle touching BC externally and AB, AC produced, 6. On the sides BC, CA, AB of the triangle ABC, respectively, pairs of points are taken, B, C1; C., A.; A., B.; such that the points of intersection of BC with B,C,, of CA with C,A,, and of AB with AB, lie in a straight line; BC, CB, intersect in L; CA, AC, in M; AB1, BA, in N. Prove that AL, BM, CN intersect in one point. 7. From the vertices of a triangle ABC, three straight lines AP, BQ, CR are drawn to pass through one point, and three straight lines AP', BQ', CR' to pass through another point, the points P, P' lying on BC, Q, Q' on CA, R, R' on AB; BQ, CR meet AP' in D ̧ D2; CR, AP meet BQ' in E1, E2; AP, BQ meet CR' in F1, F2; CD1, BD, intersect in L; AE, CE, in M; BF1, AF in N. Prove that AL, BM, CN intersect in a point. 2 2 ANHARMONIC RATIO. 19. We shall introduce, in this place, a short account of harmonic and anharmonic section, as a familiarity with this conception is useful in the higher geometrical investigations. DEF. 1. If OP, OQ, OR, OS be four straight lines intersecting in a point, the ratio sin POQ. sin ROS sin POS. sin QOR is called the anharmonic ratio of the pencil OP, OQ, OR, OS, and is expressed by the notation {O. PQRS}*. DEF. 2. If P, Q, R, S be four points in a straight line, PQ. RS the ratio is called the anharmonic ratio of the range PS. QR P, Q, R, S, and may be expressed thus [PQRS]. in using these definitions, attention must be paid to the order in which the lines or points follow one another. Thus, the anharmonic ratio of the pencil OP, OR, OQ, OS, is different from that of the pencil OP, OQ, OR, OS, the former being equal to sin POR. sin QOS sin POQ. sin ROS sin POS. sin QOR' the latter to sin POS. sin QOR' DEF. 3. If any number of straight lines, intersecting in a point, be cut by another straight line, the straight line which cuts the others is called a transversal. 20. PROP. If four given straight lines, intersecting in a point O, be cut by a transversal in the points P, Q, R, S, the anharmonic ratio of the pencil OP, OQ, OR, OS, will be equal to that of the range P, Q, R, S. *This notation is due, I believe, to Dr. Salmon. See his Conic Sections p. 273 (third edition). |