A Set of Axioms for Differential Geometry by Veblen O., Whitehead J. H.

By Veblen O., Whitehead J. H.

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State and prove the converse of Prop. V. Proposition VI. 92. In the same circle (or in equal circles) unequal chords are unequally distant from the centre and the greater chord is at the lesser distance. Statement: BD > If in the circle D, the chord EG > chord AC, BD FD FD. in and Analysis: If we can include these lines the same triangle, they can readily be compared. By Prop. and its distance from the centre can V, the chord EG FD be duplicated by a chord drawn from connection established with AC.

M, the chord OS = chord PR, Plane Geometry. 51 Fig. 44. Form right triangles with radii. Draw radii MO and MP. Then in the right MON and PMQ, OM = MP, and ON = PQ Analysis: Proof: triangles MN= MQ. '. State and prove the converse of Prop. V. Proposition VI. 92. In the same circle (or in equal circles) unequal chords are unequally distant from the centre and the greater chord is at the lesser distance. Statement: BD > If in the circle D, the chord EG > chord AC, BD FD FD. in and Analysis: If we can include these lines the same triangle, they can readily be compared.

But Add 25 /_ ; Eyz /_ yEz. ) Z xyE = ZyEz+ Z xfty, Z xyz = Z xEz = Z ABC. + or in the two triangles xyz and ABC we have two sides and (AB BC) in one = to two sides (xy and yz) in the the angle ABC between AB and BC in ABC = and other, between xy and yz in xyz, hence the two triangles xyz Then Z are equal. 52. Cor. I. Two right triangles are equal if their legs are equal. Proposition XVII. 53. In an isosceles triangle, the angles opposite the equal sides are equal ; and conversely, if in angles are equal the triangle Statement of first Z NMO triangle two of the is isosceles.

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