Elements of geometry and trigonometry . 110 GEOMETRY. PROPOSITION II. THEOREM. Any régulai polygon may he inscribed in a ciicUy and circum-scribed about one. Let ABCDE &c. be a regular poly-gon : describe a circle through the threepoints A, B, C, the centre being O, andOP the perpendicular let fall from it, tothe middle point of BC : draw AO andOD. If the quadrilateral OPCD be placedupon the quadrilateral OPBA, they willcoincide ; for the side OP is common ; P the angle OPC = OPB, each being a right angle ; hence theside PC will apply to its equal PB, and the point C will fall onB : besides, f
Elements of geometry and trigonometry . 110 GEOMETRY. PROPOSITION II. THEOREM. Any régulai polygon may he inscribed in a ciicUy and circum-scribed about one. Let ABCDE &c. be a regular poly-gon : describe a circle through the threepoints A, B, C, the centre being O, andOP the perpendicular let fall from it, tothe middle point of BC : draw AO andOD. If the quadrilateral OPCD be placedupon the quadrilateral OPBA, they willcoincide ; for the side OP is common ; P the angle OPC = OPB, each being a right angle ; hence theside PC will apply to its equal PB, and the point C will fall onB : besides, from the nature of the polygon, the angle PCD=^PBA ; hence CD will take the direction BA ; and since CD =BA, the point D will fall on A, and the two quadrilaterals willentirely coincide. The distance OD is therelore equal to AO ;and consequently the circle which passes through the threepoints A, B, C, will also pass through the point D. By thesame mode of reasoning, it miglit be shown, that the circlewhich passes through the three
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Keywords: ., bo, bookcentury1800, booksubjectgeometry, booksubjecttrigonometry
