. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. ar to AK, andprolong it to M, making NM = NP;draw MQ, and from the point R in whichit intersects AK, draw RP. The right-angled triangles RNP andRNM, have the side PN equal to NM, by construction; the side RNcommon and the included angles PNR and MNR are equal; hence,the angle NRP is equal to the angle NRM; but the angle ARQ iscqualto its opposite angle NRM; hence, the lines PR and QR makeequal angles with AK; consequently they are the required lines. Pr
. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. ar to AK, andprolong it to M, making NM = NP;draw MQ, and from the point R in whichit intersects AK, draw RP. The right-angled triangles RNP andRNM, have the side PN equal to NM, by construction; the side RNcommon and the included angles PNR and MNR are equal; hence,the angle NRP is equal to the angle NRM; but the angle ARQ iscqualto its opposite angle NRM; hence, the lines PR and QR makeequal angles with AK; consequently they are the required lines. Prop. XVI.—Show that the sum of the lines drawn from twogiven points to any point of a given line, is the least possible when theselines are equally inclined to the given line. Demonstration.—Employing the same construction as in the lastfigure, let P and Q be the given points,and PR and QR equally inclined to AK;also PS and SQ unequally inclined. The line AK is drawn through thevertex R of the triangle QRP and perpen-dicular to the bisectrix of the angle QRP; hence, from Prop. XII, Key, QR -f RP < QS -|- SP, which was to PROPOSITIONS FEOM LEGENDRE. 15
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Keywords: ., bookcentury1800, bookdecade1870, booksubjectgeometry, bookyear187
