IL, BE, CF, GD, HI, LK, AM will be equal to в C, DI, AL, AE, BF, CF, GD, HI, LK; and taking away from both CF, GD, HI, LK, there will remain AB, CD, IL, BE, AM equal to BC, DI, AL, AE, BF. But fince BE, A M are equal to AE, BF, therefore again taking these away from both, there will remain AB, CD, IL equal to BC, DI, AL.

The demonftration is much the fame, when the figure about the circle has eight, ten, &c. fides.

Therefore, &c. Which was to be demonftrated.

PROP. XII.

If from any point in the diameter of a circle be drawn two right lines to the circumference, the one perpendicular to that diameter, and the other to the middle point of the circumference of one of the femicircles upon that diameter, the fquares of these two right lines taken together, will be equal to twice the Square of the femidiameter of the circle.

Let ADB N be a circle whofe diameter is AB, and centre c, and let D be the middle point of the circumference ADB of the femicircle A DB; alfo let P be any point in the diameter A B, from which are drawn to the circum. ference the right line PM perpendicular to A B, and the right line PD to the point D: I say the fquares of PM, PD together, will be equal to twice the fquare of the femidiameter A c of the circle.

For continue out PM meeting the circumference in the point N, and join C D.

M

D

Then because MP is perpendicular to the diameter A B ; [by 3. 3.] MP, PN will be equal to one another: And therefore [by 35. 3.] the fquare of PM will be equal to the rectangle under AP, PB. But fince [by 5. 2.] the rectangle under AP, PB together with the fquare of PC is equal to the fquare of Ac; therefore will the fquares of P M, PC be equal to the fquare of A c. But becaufe of A the right angle PCD (fince the arches AD, DB are [by fuppofition] equal to one another, and fo [by

M 2

N

P C

B

27.

27. 3.] the angles AC D, D C B foo) the fquare of PD will be [by 47. 1.] equal to the fquares of PC, CD; that is, because CD, AC are equal, the square of P D will be equal to the fquares of PC and A C. Therefore [by adding equals to equals] the fquares of P M, PC, PD will be equal to twice the fquare of A C together with the fquare of PC; and taking away from both the common fquare of PC, there will remain the fquares of P M, PD equal to twice the fquare of A C.

Therefore, &c. Which was to be demonftrated.

PROP. XIII. THEOR.

In a circle if any right line be parallel to a diameter, and from the extremes of that parallel to any point in that diameter be drawn two right lines; their fquares taken together, will be equal to the fquares of the parts of the diameter taken together on each fid: the affumed point.

Let ABCD be a circle whofe diameter is A D, and centre F; let BC be any right line in the circle parallel to the diameter AD, and let E be any point taken in that diameter, from which let two right lines E B, E C be drawn to the extremities of that parallel: I fay the fquares of B E, CE taken together will be equal to the fquares of the parts A E, E D of the diameter.

For bifect BC [by 10. 1.] in the point G, and join E G,

FG, FC.

B G

A

EF

C

Then because the fide BC of the triangle BEC is bifected in G: [by prop. 3. add. 2.] the fquares of E B, EC will be equal to twice the fquare of D GC, together with twice the fquare of E G. But because BC [by fuppofition] is parallel to AD; the angles AFG, FGC [by 29. 1.] will be equal to one another. But fince B C is bifected in G, the angle FGC [by 3. 3.] will be a right angle; therefore AFG will also be a right angle: And fo [by

G

E

A

F

C

D

[by 47. 1.] the fquare of EG will be equal to the fquares of EF, FG; and twice the fquare of EG will be equal to twice the fquares of EF, F G. Again, fince FGC is a right angle, and so the fquares of F G, Gc are [by 47. 1.] equal to the fquare of F C, or to the fquare of AF (for A Fis equal to FC,) and twice the fquare of F G, and twice the fquare of G C is equal to twice the fquare of A F. Therefore, if equals be added to equals, twice the fquare of E G, twice the fquare of FG, together with twice the fquare of GC, will be equal to twice the fquare of EF, twice the fquare of FG, together with twice the fquare of AF. And taking away from both twice the fquare of F G, there will remain twice the fquare of E G, together with twice the fquare of G c, equal to twice the fquare of E F, together with twice the fquare of A F. But it has been already proved that twice the fquare of GC, together with twice the fquare of EG is equal to the fquares of EB, EC: Therefore the fquares of E B, E c will be equal to twice the fquare of EF, and twice the square of AF; that is, because AD is bifected in F; and fo [by 9. and 10. 2.] the fquares of A E, E D are equal to twice the fquares of EF, AF; the fquares of E B, E C will be equal to the fquares of A E, E D.

Therefore, &c. Which was to be demonftrated,

PROP. XIV. THEOR.

In a circle if any two right lines mutually cut one another at right angles; the four Squares of their Segments will be equal to the fquare of the diameter of the circle.

In the circle A BCE let any two right lines A C, E B mutually cut one another at right angles in the point D: I fay the four fquares of the fegments A D, DB, ED, DC, will be equal to the fquare of the diameter of the circle. For join A B, EC, AE.

Then because ADE [by fuppofition] is a right angle, and fo, [by 32. 1.] the angles A E D, E A D together are equal to one right angle, the arches AB, E C, on which thofe angles ftand, will be [by 26. 3.] together equal to

M 3

one

one half the

A

B

circumference of the circle.

C

Therefore because the angle of a femicircle is [by 31. 3.] a right angle, the fquares of AB, EC [by 47. 1.] will be equal to the fquare of the diameter of the circle. And because the angles at D are right angles [by fuppofition]; the fquares of AD, DB will be equal to the fquare of A B, and the fquares of E D, DC equal to the fquare of E c. Therefore [by adding equals to equals] the fquares of A D, DB, DE, DC will be equal to the fquares of A B, E C. But the fquares of A B, E c have been proved to be equal to the fquare of the diameter. Therefore the squares of A D, BD, ED, CD will be equal to the fquare of the diameter of the circle A BCE.

E

Therefore, &c. Which was to be demonftrated.

If from the vertex of an equilateral triangle erected upon the diameter of a circle, a right line be drawn to any point in that diameter, and from that point be drawn a right line perpendicular to the diameter meeting the circumference of the circle; the fquare of that right line, together with the fquare of that perpendicular, will be equal to the fquare of the diameter of the circle.

Let AMB be a circle whofe diameter is A B, and centre c; and let ADB be an equilateral triangle defcribed upon the diameter A B. Let p be any point taken in a B. Let the right line P D be drawn, and let PM be perpendicular to the diameter A B, meeting the circumference of the circle in M: I fay the two fquares of P D, P M will be equal to the fquare of the diameter A B of the circle.

For join c D.

Then because AC is equal to CB; AD equal to DB, and CD is common; the angles ACD, BCD [by 4. 1.] will be equal to cne another; and fo each of them will be a right angle. Therefore the fquare of A D will [by 47. 1.]

be

M

D

be equal to the two squares of AC, CD. And fince A D is equal to a B, by reafon of the equilateral triangle, and [by 4. 2.] four times the fquare of the femidiameter A c is equal to the fquare of the diameter AB: Therefore four times the fquare of A C, will be equal to the two fquares of A C, CD; and taking away the common fquare of A C from both, there will remain thrice the fquare of A c equal to the fquare of C D. Again, because [by 35. 3.] the fquare of P M is equal to the rectangle under A P, PB (for if P M be continued to meet the circumference in N, MN by 3. 3.] will be bifected in P) and fince A B is divided equally in c and unequally in P; and fo [by 5. 2.] the rectangle under A P, PB, together with the fquare of PC, is equal to the fquare of A C: Therefore the squares of PC, PM will be equal to the fquare of AC. Wherefore fince it has been proved that the fquare of C D is equal to thrice the fquare of A C, if equals be added to equals, the three fquares of C D, PC, P M will be equal to four times the fquare of A C, that is, to the fquare of AB. But becaufe [by 47. 1.] the fquare of PD is equal to the two squares of P C, C D, by adding again equals to equals, the squares of PC, PM, CD, PD will be equal to the squares of A B, PC, C D, and taking away the fquares of PC, CD from both, there will remain the fquares of PD, P M equal to the fquare of the diameter A B.

A

B

P C

N

Therefore, &c. Which was to be demonftrated.

SCHOLIUM.

There are many other elegant and useful theorems concerning the equalities of the fquares and rectangles of right lines drawn in and about a circle, a few of which I shall add at the end of the fixth book, because their demonftrations are Shorter and much eafier from the proportionality of the fides of equiangular triangles, and the equality of the rectangles under the means and extremes, than by the propofitions of the fecond book.