Let ABCD be a trapezium, whose diagonals AC, BD are bisected in F, G; join FG. The sum of the squares of AC and BD is less than the sum of the squares of the four sides, by four times the square of FG.

Since AC the base of the triangle ABC is bisected by the line BF, we have, (B. II, Prop. xii,)

AB2+BC2=2 AF2+2 BF2; and, for a similar reason,
CD2+DA2=2 AF2+2 DF2; therefore

AB2+BC2+CD2+DA2=4 AF2+2 BF2+2 DF2.

But 4 AF2=AC2.

and 2 BF2+2 DF2=4 BG2+4 FG2;

[B. II, Prop. v, Cor.]

[B. II, Prop. xII.]

therefore AB2+BC2+CD2+ DA2=AC2+4 BG2+4 FG2

=AC2+BD2+4 FG2.

(66.) THEOREM. If, from any point whatever, lines be drawn to the four corners of a parallelogram, twice the sum of their squares will be equivalent to the sum of the squares of the diagonals, increased by eight times the square of the line drawn from the given point to the intersection of the diagonals.

F

A

A

B

B

Let lines be drawn from the point P to the corners of the parallelogram ABCD, and to the intersection F of the diagonals. Then, from the triangle PDB, we have, (B. II, Prop. XII.)

PD2+PB2=2 DF2+2 PF2:

and from the triangle PAC, we have

PA2+PC2=2 AF2+2 PF2.

Hence PA2+PB2+PC2+PD2=2 DF2+2 AF2+4 PF2; consequently 2 (PA2+PB2+PC2+PD2)=BD2+AC2+8 PF2.

Cor. 1. In case the parallelogram is a rectangle, then since . .

AC=BD, it follows that the sum of the squares of the four lines drawn from the point P to the corners of the rectangle is equivalent to the square of the diagonal, together with four times the square of the line drawn from P to its middle point.

Cor. 2. Also, since, in the rectangle, we have DF=AF, it follows, from the two first equations, that the sum of the squares of the lines drawn from P to the opposite corners, is equivalent to the sum of the squares of the two lines drawn from the same point to the other two opposite corners.

Schol. If the point P is supposed to be situated at one of the corners of the parallelogram, we shall, as in the preceding general property of the parallelogram, arrive at the relation already established between the squares of the sides and the squares of the diagonals, (B. II, Prop. xiv.)

(67.) THEOREM. If, from the central point of any parallelogram, as a centre, a circle be described with any radius, the sum of the squares of the four lines drawn from any point in the circumference, to the four corners of the parallelogram, will always remain the

same.

From G the centre of the parallelogram ABCD, let a circumference of a circle be described, having any radius; also from F any point in this circumference, let lines be drawn to the four corners of the parallelogram. Then (Art. 66,) twice the sum of the squares of these four lines will be

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B

equal to the sum of the squares of the diagonals, together with eight times the square of the radius FG ; but the diagonals and the radius always remain the same, and therefore the squares of the four lines thus drawn will always amount to the same sum.

BOOK THIRD.

DEFINITIONS.

1. ANY portion of the circumference of a circle is called an arc.

2. The straight line joining the extremities of an arc is called a chord. The chord is said to subtend the arc.

3. The portion of the circle included by an arc and its chord, is called a segment.

A

B

H

Thus the space FGDF, included by the arc FGD and the chord FD, is a segment; so also is the space included by the same chord and the arc FAHBD.

4. The portion included between two radii and the intercepted arc, is called a sector.

The space BCH is a sector.

5. When a straight line touches the circumference in only one point, it is called a tangent; and the common point of the line and circumference is called the point of

contact.

6. One circle touches another, when their circumferences have only one point common.

7. A line is inscribed in a circle, when its extremities are in the circumference.

8. When a straight line cuts the circumference of a circle, it is called a secant.

9. An angle is inscribed in a circle, when its sides are inscribed.

10. A polygon is inscribed in a circle when its sides are inscribed; and under the same circumstances, the circle is said to circumscribe the polygon.

Thus AB is an inscribed line, ABC an inscribed angle, and the figure ABCDF an inscribed polygon.

11. A circle is inscribed in a polygon when its circumference touches each side, and the poly

F

gon is said to be circumscribed about the circle.

12. By an angle in a segment of a circle, is to be understood an angle whose vertex is in the arc, and whose

sides intercept the chord of said arc; and by an angle at the centre, is meant one whose vertex is at the centre. In both cases, the angles are said to be subtended by the chords or arcs which their sides include.

13. The circumference of a circle may be described by causing the extremity B of the line AB to revolve about the other extremity A, which remains fixed. In this revolution, while the line AB passes over the angular space BAC, the

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C

B

A

extremity B passes over the arc BC; and while the line passes over the angular space CAD, its extremity describes the arc CD; and so for other angles. Hence the angles at the centre are measured by the arcs inIcluded between their sides.

14. Similar arcs, similar sectors, and similar segments are such as correspond with equal angles at the centres of their respective circles.

PROPOSITION I

THEOREM. If a line, drawn from the centre of a circle, bisect a chord, it will be perpendicular to the chord; or, if the line drawn from the centre be perpendicular to the chord, it will bisect both the chord and the arc of the chord.

Let AB be any chord in a circle, and CD a line drawn from the centre C to the chord; then, if the chord be bisected at the point D, CD will be perpendicular to AB.

Drawing the two radii CA, CB,

D