# Acerca de la Geometría de Lobachevski by A. S. Smogorzhevski

By A. S. Smogorzhevski

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36). o Fig. 36. M That is, if xy and zy, then MN A line M First part: N and are the mid-points, respectively, of MN = and drawn through to ocz is || N ^ xz. to xz passes through || by Cor. I, and since two points determine a must be this line which is parallel to xz. line, straight MN Draw Second part: MN since ON through N and ON to xM, O is xO to is || parallelogram; hence MN = is || xO, but to #MNO the of xz. ) /. 70. Then, xy. || and the bases, That is, is DC is First part: MN = xO The median Cor.

Drawn tangent to the circle is perpendicular If FE perpendicular to radius is tangent to the circle A at B, then it AB. Analysis: The perpendicular point to a line. is the shortest line from a AD any other line from A to FE, cutting Since by definition the tangent to a circle in but one point the point must lie outside the Draw Proof: to the point of tangency. Statement: is a to the circle at C. touches circle, V. from it D hence AD > A to AD > AC, but AB; hence AB FE. '. AB is is AC = AB. shorter than any other line perpendicular to FE.

Six sides is polygon of seven sides is a heptagon. polygon of eight sides is an octagon, etc. may be represented schematically as follows: 3 sides. triangle quadrilateral polygons. pentagon 5 sides. hexagon 6 sides. 7 sides. heptagon ..... octagon Proposition The sum In the polygon ZC + ZD + ZE (62) times, that Analysis: XXIX. of the interior angles of a polygon is equal two right angles taken as polygon has sides. to Statement: 8 sides. 9 sides. 10 sides, etc. nonagon decagon 73. 4 sides. Since is times less two, as the ABCDEF, ZA+ZB+ + Z F = two right = eight right angles.