PROP. VI. THEOR.

If the four fides of any trapezium be bifected, and the points of bifection be joined by four right lines, the figure contained under thefe right lines will be a parallelogram, and the parallelogram will be one balf of the trapezium.

Let A B C D be a trapezium, whose fides AB, BC, CD, AD are bisected in the points E, F, G, H, and the right lines EF, FG, GH, EH join the points of bifection: I fay, the figure E F G H contained under them will be a parallelogram, and this parallelogram will be one half of the trapezium A B C D.

For draw the diagonals A C, B D.

Then fince the fides A B, B C of the triangle ABC, are bifected in the points

E, F, the right line E F

fection will [by the pre- B

joining the points of bi

cedent lemma] be pa

rallel to the diagonal AC.

In like manner the right line H G will be parallel to A C. But right lines that are parallel to the

A

D

fame right line, are [by 30. 1.] parallel to one another. Therefore the right line FE will be parallel to the right line HG. By the fame reason the right line F G will be parallel to the right line HE: Therefore the figure E F G H is a parallelogram.

Again, Let the point I be the interfection of the diagonals. Let EF, HG cut the diagonal B D in the points N, M, and the lines FG, EH the diagonal A c in the points K, L. Then because FN, G M are parallel, and cK is equal to KL; therefore [by 41. 1.] the parallelogram NK will be double to the triangle F CK. In like manner, because F G is parallel to B D, and B N is equal to N I, the parallelogram NK will be double to the triangle BF N. And fo the parallelogram N K will be equal to both the triangles BF N, FCK. So alfo will the parallelogram IG be

H 4

equal

equal to the triangles CK G, GM D. Wherefore the parallelogram N G will be equal to one half the triangle B CD. By the fame reason, the parallelogram E M will be equal to one half the triangle A B D. Wherefore the two parallelograms NG, EM together; that is, the parallelogram EFGH will be one half of the sum of the two triangles BCD, BAD; that is, one half the trapezium A B C D. Therefore, &c. Which was to be demonftrated.

If the middle points of the oppofite fides of any quadrilateral figure be joined by two right lines, the Sum of the fquares of these two right lines will be one half the fum of the fquares of the diagonals of the quadrilateral figure.

Let A B C D be a quadrilateral figure whofe fides are bifected in the points E, F, G, H, which are joined by the right lines EG, FH: I fay, the fum of the fquares of the right lines E G, F H, will be equal to one half the fum of the fquares of the diagonals A C, B D of the quadrilateral figure.

For join the right lines EF, FG, GH, EH.

F

B

C

Then because the fides of the quadrilateral figure are bifected in the points E, F, G, H [by 7. of this] the figure EFGH will be a parallelogram : And the equal oppofite fides EF, GH will be each of them one half of the diagonal A C of the quadrilateral fiG gure, and the equal oppofite fides

E

A

FG, EH each of them equal to one half the diagonal B D of the quadrilateral figure. Therefore [by 4. 2.] the fum of the fquares of the four H D fides EF, FG, GH, EH of the parallelogram E F G H will be equal to one half the fum of the fquares of the diagonals A C, B D of the quadrilateral figure ABCD. But [by 5. of this] the fum of the fquares of the diagonals EG, FH of the parallelogram; that is, of the right lines joining the middle points of the fides of the quadrilateral figure A B C D, is equal to the fum of the fquares

of

of the fides of the parallelogram. Therefore the fum of the fquares of the right lines E G, F H, will be equal to one half the fum of the fquares of the diagonals A C, B D, of the trapezium A B C D.

Therefore, &c. Which was to be demonftrated.

PROP. IX. THEOR.

If the middle points of the oppofite fides of any trapezium be joined by two right lines, the fum of the Squares of the two oppofite fides, together with twice the fquare of the right line joining the other two oppofite fides, will be equal to the fum of the fquares of the other two oppofite fides, together with twice the fquare of the right line joining the middle points of the two oppofite fides first affumed.

Let there be a trapezium A B C D, whofe fides are bifected in the points E, F, G, H. And let the right lines EG, FH, be drawn. I say, the fum of the fquares of the oppofite fides A B, CD of the trapezium, together with twice the fquare of the right line FH, will be equal to the fum of the fquares of the other two oppofite fides A D, B C, together with twice the fquare of the right line E G.

For let the point I be the interfection of the right lines EG, FH, joining the points of bifection of the oppofite fides of the trapezium; and join A 1, B I, C I, DI.

C

Then because the fide A D of the triangle A ID is bifected in the point H, the squares of A I, ID will [by 5. of this] be equal to twice the fquares of A H, HI. In like manner fince A B is bifected in E; the fquares of AI, BI will be equal to twice the fquares of AE, E 1. So likewife will the fquares of BI, IC be equal to twice the fquares of B F, FI, and the squares of CI, ID equal to twice the fquares of cG, 10. Therefore E the fquares of A I, BI, CI, DI

will be equal to twice the

fquares of AH, HI, BF, FI, A

F

B

G

1

and alfo the fquares of A 1, B I, CI, DI equal to twice

the

the fquares of AE, EI, CG, I G. Wherefore twice the fquares of AH, HI, B F, F I will be equal to twice the fquares of AE, EI, CG, IG. And [by doubling each fum] four times the fquares of AH, HI, BF, FI will be equal to four times the fquares of A E, E I, C G, I G. But [by 4. 2.] four times the fquare of AH is equal to the fquare of A D, four times the fquare of B F is equal to the fquare of BC, and four times the fquare of F I, IH is equal to twice the fquare of FH. In like manner four times the fquare of AE is equal to the fquare of A B, four times the fquare of c G equal the the fquare of c D, and four times the fquare of EI, IG equal to twice the fquare of E G. Wherefore [by 1. 2.] the fquares of A D, BC, together with twice the fquare of E G, will be equal to the fquares of ▲ B, C D, together with twice the fquare of F H.

Wherefore, &c. Which was to be demonftrated.

SCHOLIU M.

The following theorem is also easily obtained and demonfrated, viz. That in any trapezium, if the fides be bifected, and two right lines join the oppofite points of bifection, and if four right lines be drawn from their interfection to the angles of the trapezium; four times the fum of the fquares of thefe right lines will be equal to the fum of the fquares of the four fides of the trapezium, together with twice the fum of the fquares of the two right lines joining the middle paints of the fides of the trapezium.

PROP. X. THEOR.

In any five-fided right-lined figure, thrice the fum of the fquares of the fides will be equal to the fum of the jquares of the diagonals, together with four times the fum of the Iquares of the five right lines orderly joining the middle points of the diagonals.

Let there be a five-fided right-lined figure ABCDE; and let its five diagonals A G, B D, CE. DA, E B be bisected in the points F, G, H, I, K. And let each two of these middle points next to one another be joined by the five right lines FG, G H, HI, IK, K F: I fay, thrice the sum of the fquares of the fides AB, BC, CD, DE, E A of the

figure will be equal to the fum of the fquares of the diagonals A C, BD, CE, DA, EB, together with four times the fum of the fquares of the five right lines FG, GH, HI,

IK, KE.

C

For because in the trapezium A B C D, the two diagonals A C, B D'are bifected in the points F, G, and the right line FGjoins them. Therefore [by 6. of this] the fum of the fquares of A B, B C, CD, AD is equal to the fum of the fquares of AC, BD, together with four times the fquare of FG. In like manner, in the trapezium BCDE, the

D

G

H

F

K

A

E

fum of the fquares of B C, CD, DE, BE is equal to the fum of the fquares of B D, CE, together with four times the fquare of G H. And in the trapezium C DE A, the fum of the fquares of CD, DE, E A, AC is equal to the fum of the fquares of c E, A D, together with four times the fquare of HI. Alfo in the trapezium A B D E, the fum of the fquares of D E, E A, B A, B D is equal to the fum of the fquares of A D, B E, together with four times the fquare of I K. And in the trapezium E A B C, the fum of the fquares of A E, A B, BC, CE is equal to the squares of AC, B E, together with four times the fquare of K F. And adding all the former fquares together, their fum will be equal to three times the square of each fide A B, BC, CD, DE, FA together with the sum of the fquares of the five diagonals AD, BE, AC, BD, CE. Also adding all the latter fquares together, their fum will be equal to twice the fquare of each of the five diagonals A C, B D, CE, A D, BE, together with four times the fquares of the right lines FG, GH, HI, I K, K F, But fince the fum of all the former fquares is equal to the fum of all the latter; and if from both these fums be taken away the sum of the fquares of all the diagonals, there will remain thrice the fquares of each fide A B, BC, CD, DE, EA equal to the fum of the fquares of the diagonals AC, BD, CE, AD, BE, together with four times the

fquares