Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre (1875)
Legendre, A. M. (Adrien Marie), 1752-1833, Davies, Charles, 1798-1876 · New York, Chicago [etc.] A.S. Barnes & company · leaf 51
nes are drawn from any point P to the fourvertices of a rectangle, show that the sum of the squares of the twolines drawn to the extremities of one diagonal is equal to the sum ofthe squares of the two lines drawn to the extremities of the otherdiagonal. Demonstration.—Let ABDC be any rectangle,… … F B PC2 = PF2 + FC2, whence, by addition, PB + PC = PE + PF + EB + FC . . . (1). From the right-angled triangles PEA and PFD, we have, PA2 = PE2 + EA2 PD2 = PF2 + FD2,whence, by addition, PA2 + PD2 = PE2 + PF2 + FD2 + EA2 . . . (2). PROPOSITIONS FROM LEGENDRE. 45 Because, EB = FD, and EA == FC, the…
Geometry
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