An elementary treatise on modern pure geometry (1893)
Lachlan, Robert · London, New York, Macmillan and Co. · leaf 75
asthe transversal XTZ. 105. If X,Y, Z are points on the sides of a triangle ABCsuch that BX CY AZ_CX AY BZ~ the points X, Y, Z are collinear. Let the line joining the points Y and Z cut BG in the pointX. By the last article, we have BX CY^ 4?=iGX-AYBZHence, we must have BX:CX = BX:CX.Therefore X… … This relation is easily deduced from that given in § 104, for wehave BX-.CX^AB.sinBAX :^C.sinCAX, CY: AY=BG . sin GBY : BA .sin^BF,AZ:BZ = CA. sin ACZ : CB . sin BCZ. A TRIANGLE. 59 Hence, BX CY AZ_smBAX sin GBY sin AGZaXAYBZ sin GAX sin ABY sin BCZ But by § 104, the former product is equal to…
Geometry
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