PROP. XXXVI. (LII.) To bisect a parallelogram by a straight line drawn through a given point in one of its sides. (LIII.) A trapezium, which has two of its sides parallel, is the half of a rectangle between the same parallels, and having its base equal to the aggregate of the two parallel sides of the trapezium. PROP. XXXVII. (LIV.) A plane rectilineal figure of any number of sides being given, to find an equal rectilineal figure, which shall have the number of its sides less, or greater, by one, than that of the given figure. (LV.) Hence, first, to find a triangle, which shall be equal to any given plane rectilineal figure: secondly, to find a polygon of any given number of sides which shall be equal to a given triangle. PROP. XXXVIII. (LVI.) The diameters of any parallelogram divide it into four equal triangles. (LVII.) Of all triangles, which are between the same parallels, that which stands on the greatest base is the greatest. (LVIII.) The straight line, joining the vertex and the bisection of the base of any triangle, bisects every other straight line that is parallel to the base and is terminated by the two remaining sides of the triangle. (LIX.) Hence, if two opposite sides of a trapezium be parallel to one another, the straight line, joining their bisections, bisects the trapezium. (LX.) To bisect a given trapezium by a straight line drawn from any of its angles. (LXI.) To bisect a given triangle, by a straight line drawn through a given point in any one of its sides. PROP. XL. (LXII.) Equal triangles, which have their bases in the same straight line and which are between the same parallels, stand upon equal bases. PROP. XLI. (LXIII.) To describe a parallelogram, the area and perimeter of which shall be respectively equal to the area and perimeter of a given triangle. (LXIV.) The two triangles formed by drawing straight lines, from any point within a parallelogram, to the extremities of either pair of opposite sides, are, together, half of the parallelogram. (LXV.) If two sides of a trapezium be parallel, the triangle contained by either of the other sides, and the two straight lines drawn from its extremities to the bisection of the opposite side, is the half of the trapezium. (LXVI.) The triangle contained by the straight lines joining the points of the bisection of the three sides of a given triangle, is one-fourth part of the given triangle, and is equiangular with it. (LXVII.) Hence, if the four sides of any given quadrilateral rectilineal figure be bisected, the figure contained by the straight lines joining the several points of the bisection, shall be a parallelogram, which is the half of the given figure; also the four sides of this parallelogram shall be, together, equal to the two diagonals of the given figure. COR. It is manifest that the straight lines, which join the opposite points of bisection of the sides of any trapezium, bisect each other. PROP. XLIII. (LXVIII.) To describe a parallelogram, which shall be of a given altitude, and equiangular with, and also equal to, a given parallelogram. COR. Hence, a rectangle may very readily be found, which shall be equal to a given square, and shall have one of its sides equal to a given straight line. PROP. XLV. (LXIX.) If there be any number of rectilineal figures, of which the first is greater than the second, the second than the third, and so on, the first of them shall be equal to the last together with the aggregate of all the differences of the figures. (LXX.) To find a rectangle, which shall have one of its sides equal to a given finite straight line, and which shall be equal to the excess of the greater of two given rectilineal figures above the less. SCHOLIUM. From the twenty-second, and the forty-fifth propositions, of this first book of Euclid, may be deduced a method of surveying, planning, and measuring irregular plots of ground, which have |