33. If the rectangle contained by the diagonals of a quadrilateral he equal to the sum of the rectangles con-tained by the opposite sides, a circle can be descrihed round the quadrilateral.
This is the converse of VI. D; it can be demonstrated indirectly with the aid of 32.
34. It is required to find a point in a given straight line, such that the rectangle contained hy its distances from two given points in the straight line may he equal to the rectangle contained by its distances from two other given points in the straight line.
Let A, B, C, D be four given points in the same straight line: it is required to find a point in the straight
line, such that the rectangle contained by its distances from A and B may be equal to the rectangle contained by its distances from C and D.
On AD describe any triangle AED; and on CB describe a similar triangle CFB, so that CF is parallel to AE, and BF to DE; join EF, and let it meet the given straight line at O, Then shall be the required point.
For, OE is to OA as OE is to OC (VI. 4); therefore OE is to OF as OA is to OC (V. 16). Similarly OE is to OF as OD is to OB. Therefore OA is to OC as OD is to OB (V. 11). Therefore the rectangle OA, OB is equal to the rectangle OC, OD.
