. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. Prop. XLIII.—Let an equilateral triangle be inscribed in a circle,and let two of the subtended arcs be bisected by a chord: then showthat the sides of the triangle divide the chord into three equal parts. Demonstration.—Let ABC be an equilateral triangle inscribed ina circle ASBCP, and let P and S be the middle points of the arcsAPC and ASB. PROPOSITIONS FROM LEGENDRE. 31 Draw PS, cutting AC and AB in Q and R; also draw AP andAS. The angles ASR and RAS
. Selected propositions in geometrical constructions and applications of algebra to geometry. Being a key to the appendix of Davies' Legendre. Prop. XLIII.—Let an equilateral triangle be inscribed in a circle,and let two of the subtended arcs be bisected by a chord: then showthat the sides of the triangle divide the chord into three equal parts. Demonstration.—Let ABC be an equilateral triangle inscribed ina circle ASBCP, and let P and S be the middle points of the arcsAPC and ASB. PROPOSITIONS FROM LEGENDRE. 31 Draw PS, cutting AC and AB in Q and R; also draw AP andAS. The angles ASR and RAS are equal be-cause they are measured by halves of theequal arcs AP and SB; hence, AR = like manner it may be shown that AQ —PQ. In the triangle AQR, the angle ARQis measured by \ (SB -f PA), the angleAQR is measured by -J- (AS -f PC), and the angle QAR is measuredby JCB, hence, the three angles are equal, and consequently QR =AQ = AR, or PQ = QR = RS, which was to be Prop. XLIV.—Find a point within a triangle, such that the anglesformed by drawing lines from it to the three vertices of the triangle,shall be equal. Solution.—Let ABD be the given triangle. On AB constructan arc that will contain an angle equal tofour-thirds of a right angle; on BD con-struct another arc that will contain anangle equal to four-thirds of a right angle,and intersecting the first at O; draw OD,OA, and OB. The angles AOD and DOB are eachequal to four-thirds of a right angle, and consequently, the angle AOBis also equal to four-thirds of a right angle; hence, O is the requiredpoint. Prop. XLV.—Inscribe a circle in a quadrant of a given —Let CBPD be the given quadrant.
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Keywords: ., bookcentury1800, bookdecade1870, booksubjectgeometry, bookyear187
