Elements of geometry and trigonometry . 33 •- Scholium, If one of the sides were greater than the sum ofthe other two, the arcs would not intersect each other : but thesolution will always be possible, when the sum of two sides, any-how taken, is greater than the third. 62 GEOMETRY. PROBLEM XI. Two sides of a triangle, and the angle opposite one ofthe?n, beinggiven, to describe the triangle. Let A and B be the given sides, and C the given are tw^o cases. Fiy^st. When the angle C is a rightangle, or when it is obtuse, makethe angle EDF=C; take DE=:A ;from the point E as a centr
Elements of geometry and trigonometry . 33 •- Scholium, If one of the sides were greater than the sum ofthe other two, the arcs would not intersect each other : but thesolution will always be possible, when the sum of two sides, any-how taken, is greater than the third. 62 GEOMETRY. PROBLEM XI. Two sides of a triangle, and the angle opposite one ofthe?n, beinggiven, to describe the triangle. Let A and B be the given sides, and C the given are tw^o cases. Fiy^st. When the angle C is a rightangle, or when it is obtuse, makethe angle EDF=C; take DE=:A ;from the point E as a centre,with a radius equal to the givenside B, describe an arc cutting DFin F; draw EF : then DEF will bethe triangle required. In this first case, the side B mustbe greater than A ; for the angle C,being a right angle, or an obtuse an-gle, is the greatest angle of the tri-angle, and the side opposite to it must, therefore, also be thegreatest (Book I. Prop. XIII.).. Secondly, If the angle C isacute, and B greater than A, thesame construction will again ap-ply, and DEF will be the trianglerequired. But if the angle C is acute, andthe side B less than A, then thearc described from the centre E,with the radius EF=B, will cutthe side DF in two points F andG, lying on the same side of D :hence there will be two trianglesDEF, DEG, either of which willsatisfy the conditions of the pro-blem. 33 j-
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