# Calendar: 1854-1855 Page 407

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the school 409 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 art 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 To describe rectilineal figure which shall be similar to one and equal to one another given rectilineal figure The least chord of circle that can be drawn through given point is perpendicular to the diameter passing through the same point 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 other two sides Three spectators on level plain and not in the same straight line observe bird rise from the ground and fly upwards supposing the bird to be always at equal distances from the three spectators determine the point from which it rose and the direction of its path 10 Λ Β is the diameter of circle being the centre and Ε are two points in at the same distances from Ρ is any chord through join DP show that DP2 Q- Ρ Q2 is constant 11 Of the two squares which can be inscribed in right angled triangle which is the greatest 12 Given the chords of two arcs in known circle find the chord of the difference of the arcs

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