DIAGONAL

, is a right line drawn across a figure, from one angle to another; and is sometimes called a diameter. It is used chiefly in quadrilateral sigures, viz, in parallelograms and trapeziums.

1. Every diagonal, as AC, di- vides a parallelogram into two equal parts or triangles ABC, ADC.

2. Two diagonals, AC, BD, drawn in a parallelogram, do mutually bisect each other; as in the point E. |

3. Any line, as FG, drawn through the middle of the diagonal of a parallelogram, is bisected by it at the point E; and it divides the parallelogram into two equal parts, BFGC and AFGD.

4. The diagonal of a square is incommensurable with its side.

5. In any parallelogram, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals.

6. In any trapezium, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals together with 4 times the square of the distance between the middle points of the diagonals.

7. In any trapezium, the sum of the squares of the two diagonals is double the sum of the squares of two lines bisecting the two pairs of opposite sides.

8. In any quadrilateral inscribed in a circle, the rectangle of the two diagonals is equal to the sum of the two rectangles under the two pairs of opposite sides.

Diagonal Scale. See Scales.