. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. 68 KEY. clearing of fractions and transposing, we have2x2 + 2y2 -z2 = 4a2 •? • • (1) 2x2 +? 2z2 - y< 4b2 • • (2) 2y2 + 2z2 -x2 -4c2 . • . (3);subtracting equation (2) from (1), we have multiplying equation (3) by 2, and adding to (2) 3y2 + 6z2 - 8c2 + 462 •subtracting equation (4) from (5), 9z2 = 8c2 -f 4b2 - 4a2 + 462 ;. z = ± f y/2b2 + 2c2 - -a2 X z= ±jv^ + 262 - -c2 y = ± J v/2a^ + 2c2 — 62. Obr. Through G and E, the middle points of AC and BC, dr
. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. 68 KEY. clearing of fractions and transposing, we have2x2 + 2y2 -z2 = 4a2 •? • • (1) 2x2 +? 2z2 - y< 4b2 • • (2) 2y2 + 2z2 -x2 -4c2 . • . (3);subtracting equation (2) from (1), we have multiplying equation (3) by 2, and adding to (2) 3y2 + 6z2 - 8c2 + 462 •subtracting equation (4) from (5), 9z2 = 8c2 -f 4b2 - 4a2 + 462 ;. z = ± f y/2b2 + 2c2 - -a2 X z= ±jv^ + 262 - -c2 y = ± J v/2a^ + 2c2 — 62. Obr. Through G and E, the middle points of AC and BC, drawGE; then will GE be parallel to the base AB (Bk. IV^. Prop. 16) :and since AC is equal to twice CG, AD will be equal to twice GP,or its equal PE. But the triangles AOD and POE are similar; and, since AD isequal to twice PE, AO is equal to twice OE ; that is, If three lines be drawn from the vertices of the three angles of atriangle to the middle points of the opposite sides, the distance fromeither vertex to the point of intersection, will be two-thirds of thebisecting line. Prop. LXXXVIII.—In a, triangle, having given the three sides, tofind the radius of the inscribed circle. Let ABC be a triangle, CD perpendicular to the base, O thecentre, and OP the radius of the inscribed circle. PROBLEMS FROM LEGENDRE. 69 Let BC = a, AB = c, AC = 6, OP = r, AD = z, CD = = 62 _ z% . . (i)y #2 = a2 — (c — zf; whence,S2 — z2 = a2 — c2 + 2cz — 22
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Keywords: ., bookcentury1800, bookdecade1870, booksubjectgeometry, bookyear187
