PROP. XV. PROB. To inscribe an equilateral and equiangular hexagon in a given circle. To inscribe an equilateral and equiangular quindecagon in a given circle. A B [53] BOOK VI. PROP. I. THEOR. Triangles and parallelograms having the same altitude, are to one another as their bases. KAL HGFCB PE D M N COR.-Hence it is manifest that triangles and parallelograms, having equal altitudes, are proportional to their bases. PROP. II. THEOR. If a straight line be drawn parallel to one of the sides of a triangle, it will cut the other sides, or the other sides produced, proportionally: and if the sides, or the sides produced, be cut proportionally, the straight line which joins the points of section, will be parallel to the remaining side of the triangle. PROP. III. THEOR. If an angle of a triangle be bisected by a straight line which also cuts the base, the segments of the base shall be proportional to the other sides of the triangle; or, if the segments of the base be proportional to the other sides of the triangle, the straight line drawn from the vertex to the point of section, shall bisect the vertical angle. B E COR.—If a line, bisecting an external angle of a triangle, meet the opposite side produced, that whole produced side, and its external segment, will be proportional to the sides which contain the angle adjacent to the external bisected angle. F E D PROP. IV. THEOR. The sides about the angles of equiangular triangles are proportional; and those which are opposite to the equal angles are homologous sides, that is, are the antecedents or consequents of the ratios. B A F PROP. V. THEOR. If the sides of two triangles be proportional, the triangles shall be equiangular, and have their equal angles opposite to homologous sides. |