. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. S : OP . . (1). In like manner, we have from the triangles BQR and BOA, the proportion, BQ : BO : : QR : OA . . (2). Because the first three terms of proportions (1) and (2) are equal,each to each, their fourth terms must be equal, that is, OP = , the circle whose radius is OA, passes through P. The distanceOC is equal to OA (Prop. IV, Key); hence, the circle whose centreis O passes through C. Furthermore, BA is perpendicular to OA, atits extrem
. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. S : OP . . (1). In like manner, we have from the triangles BQR and BOA, the proportion, BQ : BO : : QR : OA . . (2). Because the first three terms of proportions (1) and (2) are equal,each to each, their fourth terms must be equal, that is, OP = , the circle whose radius is OA, passes through P. The distanceOC is equal to OA (Prop. IV, Key); hence, the circle whose centreis O passes through C. Furthermore, BA is perpendicular to OA, atits extremity, and BC is perpendicular to OC, at its extremity;hence, the circle whose radius is OA, is tangent to both sides of thegiven angle; it is therefore the required circle. Prop. XLVII.—Show that the middle points of the sides of anyquadrilateral are the vertices of an inscribed parallelogram. Demonstration.—Let ABCD be any quadrilateral, and let P, Q,R, and S, be the middle points of its foursides. Draw PQ, QR, RS, and SP; also drawthe diagonals BD and AC. Because AQ = QB, and AP = PD,we have the proportion, AP : PD : : AQ : 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 proved.
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Keywords: ., bookcentury1800, bookdecade1870, booksubjectgeometry, bookyear187
