# Calendar: 1854-1855 Page 355

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general literature and science 357 JUNIOR MATHEMATICAL SCHOLARSHIP £uclfa anil Citgonomrtn To describe parallelogram equal to given rectilineal figure and having an angle equal to given rectilineal angle If straight line be bisected and produced to any point the square of the whole line thus produced and the square of the part of it produced are together double of the square of half the line bisected and of the square of the line made up of the half and the part produced If two straight lines cut one another within circle the rectangle contained by the segments of one of them is equal to the rectangle contained by the segments of the other To inscribe circle in given equilateral and equiangular pentagon Similar polygons may be divided into the same number of similar triangles having the same ratio to one another that the polygons have and the polygons have to one another the duplicate ratio of that which their homologous sides have If straight line stand at right angles to each of two straight lines in the point of their intersection it shall also be at right angles to the plane which passes through them that is to the plane in which they are Construct an octahedron on given line is point in the diameter AB of circle of which the circumference is divided into 2η equal parts show that the sum of the squares of the lines from to the divisions of the circum- ference ב η SA SB2 Of the three squares which can be inscribed in an acute- angled triangle the greatest is that which has two angles in the least side 10 The sum of the squares of the diagonals of trapezoid is equal to twice the rectangle of the parallel sides together with the squares of the two other sides

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