Elements of geometry and trigonometry . Cor. Hence the triangle ABC may be described by meansof DEF, as DEF is described by means of ABC. Triangles80 described are called polar triangles, or supplemental tri-angles. BOOK IX. 193 PROPOSITION VIII. THEOREM. The same supposition continuing as in the last Proposition, eachangle in one of the triangles, will be measured by a semicir-aimference, minus the side lying opposite to it in the othertriaji[ For, produce the sides AB,AC, if necessary, till they meetEF, in G and l\. Tiie point Abeing the pole of the arc GH,the angle A will be measuredby
Elements of geometry and trigonometry . Cor. Hence the triangle ABC may be described by meansof DEF, as DEF is described by means of ABC. Triangles80 described are called polar triangles, or supplemental tri-angles. BOOK IX. 193 PROPOSITION VIII. THEOREM. The same supposition continuing as in the last Proposition, eachangle in one of the triangles, will be measured by a semicir-aimference, minus the side lying opposite to it in the othertriaji[ For, produce the sides AB,AC, if necessary, till they meetEF, in G and l\. Tiie point Abeing the pole of the arc GH,the angle A will be measuredby that arc (Prop. VI.). Butthe arc EH is a (juadrant, andlikewise GF, E being the poleof All. and F of AG ; henceEH-rGF is equal to a semi-circumference. Now, EH-hGF is the same as EF+GH ; hence the arc GH, which mea-sures the angle A, is equal to a semicircumfcrence minus theSide EF. In like manner, the angle B will be measured by« c/rc—DF : the angle C, by J- ciVc—DE. And this property must be reciprocal in the two triangles,smce each of them is described in a similar manner by meansof the other. Thus we shall find the angles D,E,F, of the triangleBEFtobe measured respectivelyby ^ r/rc—BC, ^ arc—AC,-I ciVc—AB. Thus the angle D, for example, is measured bythe arc MI ; but MI-fBC = MC + BI=A drc. ; hence the arcMI, the measure of D, is ecjual to -h circ.—BC ; and so of allthe rest. Scholium. It must further be observed,that besid
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