Elements of geometry and trigonometry . PROBLEM XV. To inscribe a circle in a. given tîiangle. Let ABC be the given triangle. Bisect the angles A and B, bythe lines AO and BO, meeting inthe point O ; from the point O,let fall the perpendiculars OD,OE, OF, on the three sides of thetriangle: these perpendiculars willall be equal. For, by construe-. BOOK III. 69 lion, we have the angle DAO = OAF, the right angle ADO =AFO ; hence the third angle AOD is equal to the tiiird AOF(Book I. Prop. XXV. Cor. 2.). INIoreover, the side AO is com-mon to the two triangles AOD, AOF ; and the angles adjacentto t
Elements of geometry and trigonometry . PROBLEM XV. To inscribe a circle in a. given tîiangle. Let ABC be the given triangle. Bisect the angles A and B, bythe lines AO and BO, meeting inthe point O ; from the point O,let fall the perpendiculars OD,OE, OF, on the three sides of thetriangle: these perpendiculars willall be equal. For, by construe-. BOOK III. 69 lion, we have the angle DAO = OAF, the right angle ADO =AFO ; hence the third angle AOD is equal to the tiiird AOF(Book I. Prop. XXV. Cor. 2.). INIoreover, the side AO is com-mon to the two triangles AOD, AOF ; and the angles adjacentto the equal side are ecjual : hence the triangles themselves areequal (Book 1. Prop. VI.) ; and DO is equal to OF. In the samemanner it may be shown that the two triangles BOD, BOE,are equal ; therefore OD is equal to OE ; therefore the threeperpendiculars OD, OE, OF, are all equal. Now, if from the poin-t O as a centre, with the radius OD,a circle be described, this circle will evidently be inscribed inthe triangle ABC ; for the side AB, being perpendicular to theradius at its extremity, is a tangent ; and the same thing is trueof the sides BC, AC. Scholium. The three lines which bisect the angles of a tri-angle meet in the same point. PROBLEM XVI. On a given straight line to describe a segment that shall containa given angle ; that is to s
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