This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1890 Excerpt: ...TRGQ is harmonic, since TP, PG bisect the angle QPR. Hence PG is the polar of T. Hence the pole of QR lies on PG since the pole of PG lies on QR. 34. If the tangents at P and Q meet in T and TA meet PQ in L, the range DPLQ is harmonic; hence the pencil TD, TP, TL, TQ and the range DBAO are harmonic. Therefore ABDC is a ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1890 Excerpt: ...TRGQ is harmonic, since TP, PG bisect the angle QPR. Hence PG is the polar of T. Hence the pole of QR lies on PG since the pole of PG lies on QR. 34. If the tangents at P and Q meet in T and TA meet PQ in L, the range DPLQ is harmonic; hence the pencil TD, TP, TL, TQ and the range DBAO are harmonic. Therefore ABDC is a harmonic range. 35. If the pencil joining BPAQ to any point on the curve is harmonic, the pencil formed by joining them to any other point on the conic is harmonic. For if BK, PK, AK, QK meet the directrix in bpaq, bpaq is a harmonic range, provided KEBPAQ be a harmonic pencil. And the angles bSp, pSa, aSq, qSb are half the angles BSP, PSA, ASQ, QSB; Hence the pencil Sb, Sp, Sa, Sq is the same wherever A' be taken on the curve. Now PQ goes through O the pole of AB: let PQ meet AB in R. Then if T be the pole of PQ, TARB is a harmonic range. Therefore the pencil joining Q to BPAQ is harmonic; hence the pencil joining q to BPAQ is harmonic. Hence Pq bisects AB since AB, qQ are parallel. 36. Four circles can be described so as to touch the sides of a triangle, and the reciprocal of the radius of the inscribed circle is equal to the sum of the reciprocals of the radii of the other three. If the triangle be equilateral the inscribed circle touches the three escribed circles. 37. If the tangents at P and Q meet the axes in T and V; the angle PSQ = SQV-SPT=SFQ-STP-= VST. If SW be perpendicular to P'Q', the tangents at the vertices intersect in W. Draw SYZ perpendicular to the tangents at P and Q. Then WSP', WSQ' are supplementary to WYP', WZQ'. Hence P'SQ/, PSQ are supplementary. If two circles intersect in P, Q the angle between the tangent at P, Q is equal to the angles which the centres subtend at S and supplementary to the angle which PQ subtends...
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