. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. 34 KEY. Hence, from Bk. IV, Prop. 16, PQ is parallel to DB. In like mannerit may be shown that RS is parallel to DB: hence, PQ and SR areparallel to each other. The lines QR and PS are parallel to AC, forsimilar reasons; hence, they are parallel to each other, and conse-quently, the figure PQRS is a parallelogram inscribed in the givenquadrilateral, which was to be Q M Prop. XLVIII.—Inscribe in a given triangle, a triangle whose sidesshall be p
. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. 34 KEY. Hence, from Bk. IV, Prop. 16, PQ is parallel to DB. In like mannerit may be shown that RS is parallel to DB: hence, PQ and SR areparallel to each other. The lines QR and PS are parallel to AC, forsimilar reasons; hence, they are parallel to each other, and conse-quently, the figure PQRS is a parallelogram inscribed in the givenquadrilateral, which was to be Q M Prop. XLVIII.—Inscribe in a given triangle, a triangle whose sidesshall be parallel to the sides of a second given triangle. Solution.—Let ABC be the first given triangle, and let DEF bethe second given triangle. Take any point P, on the side AB, andthrough it draw PQ parallel to DE; fromQ draw QR parallel to EF, and from Pdraw PR parallel to DF, intersecting PR inR; draw BR and prolong it to S. ThroughS draw SM parallel to RQ, and SN parallelto RP, and then join M and N. From the triangles BSN and BRP,we have (Bk. IV, Prop. 15), BR : RS : : BP : PN . . (1).From the triangles BSM and BRQ, we have, like reasons, BR : RS : : BQ : QM . . (2).Combining proportions (1) and (2), we have,BP : PN : : BQ : QM;hence, from (Bk. IV. Prop. 16), we have, NM parallel to PQ. Thesides of the inscribed triangle NSM,-are therefore parallel to the sidesof the triangle PRQ, and consequently, to the sides of the triangleDFE. The triangle NSM is therefore the required triangle. PROPOSITIONS FROM
Size: 1693px × 1476px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1800, bookdecade1870, booksubjectgeometry, bookyear187
