Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/373

 angle FDE is equal to the angle BAC. Shew also that AFDE is half the triangle ABC.

114. Two triangles of equal area stand on the same base and on opposite sides: show that the straight line joining their vertices is bisected by the base or the base produced.

115. Three parallelograms which are equal in all respects are placed with their equal bases in the same straight line and contiguous; the extremities of the base of the first are joined with the extremities of the side opposite to the base of the third, towards the same parts: shew that the portion of the new parallelogram cut off by the second is one half the area of any one of them.

116. ABCD is a parallelogram; from D draw any straight line DFG meeting BC at F and AB produced at G; draw AF and CG shew that the triangles ABF, CFG are equal,

117. ABC' a given triangle: construct a triangle of equal area, having for its base a given straight line AD coinciding in position with 'AB.

118. ABC is a given triangle: construct a triangle of equal area, having its vertex at a given point in BC and its base in the same straight line as AB.

119. ABCD is a given quadrilateral: construct another quadrilateral of equal area having AB for one side, and for another a straight line drawn through a given point in CD parallel to AB.

120. ABCD is a quadrilateral: construct a triangle whose base shall be in the same straight line as AB, vertex at a given point P in CD, and area equal to that of the given quadrilateral.

121. ABC is a given triangle: construct a triangle of equal area, having its base in the same straight line as AB, and its vertex in a given straight line parallel to AB.

122. Bisect a given triangle by a straight line drawn through a given point in a side.

123. Bisect a given quadrilateral by a straight lino drawn through a given angular point.

124. If through the point O within a parallelogram ABCD two straight lines are drawn parallel to the sides, and the parallelograms OB and OD are equal, the point O is in the diagonal AC.