< KBF=2 KBG, .. (E. 26. 1.) FB=GB. If, .., from F and G, as centres, at the equal distances FB, GB, two equal circles be described, they will pass (constr. and S. 3. 1. cor. 2.) the one through A, and the other through B, and (S. 6. 3.) they will touch one another in the point B.

9. PROBLEM.

PROP. VIII.

To draw a tangent to a circle, which shall be parallel to a given finite straight line.

Let ABC be a given circle, and XY a given

straight line: It is required to draw a straight line which shall touch the circle ABC, and which shall be parallel to XY.

Find (E. 1. 8.) the centre K of the circle ABC; from K draw (E. 12. 1.) the diameter AKC to XY; and from either of the extremities, as C, of AC draw (E. 11. 1.) ZCW 1 to AC.

Then since ZCW is 1 to AC, at its extremity

C, it touches (E. 16. 3. cor.) the circle ABC:

And since (constr.) the two

two right angles,
angles, .. (E. 28. 1.)

XY.

XDC, DCZ are

ZW is parallel to

10. COR. Hence a tangent may be drawn to a circle which shall make with a given straight line an equal to a given rectilineal angle..

For let it be required to draw a tangent to the circle ABC, which shall make with a given straight line VW an equal to a given : Take any point Y in VW, and at the point Y, in VY, make (E. 23. 1.) the 4 VYX equal to the given : If, then, the tangent ZW be drawn (S. 8. 3.) parallel to YX, it will make (E. 29. 1.) the

ZWV = ZXYV, which is equal to the given 4.

PROP. IX.

11. PROBLEM. The diameter of a circle having been produced to a given point, to find in the part produced, a point from which, if a tangent be drawn to the circle, it shall be equal to the segment of the part produced, that is between the given point and the point found.

Let the diameter AB of the circle ABC be produced to the given point D: It is required to find in BD a point from which if a tangent be drawn to the circle, it shall be equal to the part of BD which is between that point and D.

F

B

E

K

Find the centre K of the circle ABC; from K draw (E. 11. 1.) KC 1 to AB; join D, C, and let DC meet the circumference in E; join K, E; from E draw EF to KE and let EF meet BD in F: Then is F the point which was to be found.

For (E. 13. 1.) the KEC, KEF, FED are together equal to two right angles; as are, also, (E. 32. 1.) the three / DCK, CKD, and KDC, of the A DKC: But since (E. 15. def. 1.) KE=KC, .. (E. 5. 1.) the 2 KEC= KCE; and (constr.) the KEF, CKD are equal, each of them being a right angle; .. the remaining 4 FED is equal to the remaining 4 KDC or FDE: .. (E. 6. 1.) FE FD; and since (constr.) EF is perpendicular to the semi-diameter KE, at its extremity E, .. (E. 16. 3.) FE touches the circle ABC. Q.E.F.

PROP. X.

12. PROBLEM. To describe a circle which shall have a given semi-diameter and its centre in a given straight line, and shall also touch another straight line, inclined at a given angle to the former.

Let AX and AY be two given straight lines in

X

D

C

Y

B

clined to one another at a given angle; and let L be a given finite straight line: It is required to describe a circle, which shall have its centre in AY, and its semi-diameter equal to L, and which shall touch AX.

From the point A, in AX, draw (E. 11. 1.) AB 1 to AX, and make AB = L; through B draw (E. 31. 1.) BC parallel to AX, and through C draw CD parallel to AB: Wherefore, DB is a □; .. (E. 34. 1.) DC=AB; and since (constr.) the ▲ BAD is a right ≤; .. (E. 29. 1.) the ADC is also a right : It is manifest, ..., that a circle described from C as a centre, at the distance CD, will (E. 16. 3. cor.) touch AX; and its semi-dia

meter CD has been shewn to be equal to AB, which was made equal to the given straight line L. Q. E. F.

PROP. XI.

13. PROBLEM. To describe a circle, the circumference of which shall pass through a given point, and touch a given straight line in another given point.

Let B be a given point in the given straight

line XY, and let A be any other given point, without that line: It is required to describe a circle the circumference of which shall pass through A and touch XY in B.

From B draw (E. 11. 1.) BC 1 to XY; join A, B; bisect (E. 10. 1.) AB in D, and from D draw DK 1 to AB; .. (S. 3. 1. cor. 2.) K is equidistant from A and B: It is manifest, therefore, that the circumference of a circle described from K as a centre, at the distance KB will pass through