vertical equal to a given 4, and its two remaining sides in a given ratio to one another.

Upon EF describe (E. 33. 3.) a segment of a circle EKF, capable of containing an equal to the given 4, and complete the circle EKFG; divide (E. 10. 6.) EF in H, so that EH is to HF in the given ratio; bisect (E. 30. 3.) ÉGF in G; draw GH, and produce it to meet the circumference in K; lastly, join E, K, and F, K: Then is EKF the A which was to be constructed.

For since (constr.) EG=FG, .. (E. 27. 3.) the EKG = FKG, so that the EKF is bisected by KH;

.. (E. 3. 6.) KE:KF::EH:HF: That is (constr.) KE is to KF in the given ratio, and the vertical EKF is equal to the given angle.

PROP. XXVI.

34. PROBLEM. A given finite straight line being divided into any two given parts, to divide it again, so that the rectangle contained by the two former given parts shall have a given ratio to the rectangle contained by the two latter parts.

Describe (S. 21. 6.) a square which shall be to the rectangle, contained by the given parts of the given line, in the given ratio; and divide (S. 71. 3.)

the given line into two parts, so that the rectangle contained by them shall be equal to the square so described: It is manifest that this rectangle will be to the rectangle, contained by the two given parts, in the given ratio.

PROP. XXVII.

35. PROBLEM. To draw a straight line to touch a given arch of a circle, so that being terminated by the semi-diameters, produced, which bound the arch, it shall be divided by the point of contact, into two parts that are to one another in a given ratio.

Let LBM be a given arch of the circle ALBM,

terminated by the two semi-diameters KL and KM: It is required to draw a tangent to the circle, so that, being terminated by KL and KM, produced, it shall be divided, by the point of its

R

contact, into two segments, that are to one another in a given ratio.

Take any straight line CD, and divide it (E. 10. 6.) in H in the given ratio; draw (E. 11. 1.) HE to CD, and let HE be cut in E, by a segment of a circle described (E. 33. 3.) upon CD, capable of containing an equal to the given and D, E; .. the

2 LKM; and join C, E, CED = 4 LKM; lastly, draw (S. 8. 3. cor.) FBG touching the circle ALBM, and making with KL, produced, an 4 KFG ECD: Then is the tangent FG divided in B, so that FB is to BG in the given ratio.

For join K, B; .. (constr. and E. 18. 3.) the at B are right; as are, also, the at H; and (constr.) the ECH = 4 KFB; .. (S. 26. 1.) the AECH, KFB are equiangular; and since the < CEH=2 FKB, and that (constr.) the whole CED whole/ FKG, .. the HED BKG, and (S. 26. 1.) the EHD, KBG are equiangular, as are, likewise, the CED, FKG;

.. (E. 4. 6.)

CH: HE::FB:BK; and HE: HD::BK: BG;

.. (E. 22. 5.) CH: HD:: FB: BG:

But (constr.) CH is to HD in the given ratio; FB is to BG in the given ratio.

36. PROBLEM.

PROP. XXVIII.

Two points being given, one in each of two parallel straight lines, and a third point being also given, without them, to draw, from that third point, a straight line so to cut the parallels, as that the segments of the parallels, between it and the two first points, shall be to one another in a given ratio.

straight lines; A and B the two given points in them; and Ca given point without them: It is required to draw from C a straight line cutting PQ and RS, so that the segments of PQ and RS, between the cutting line and the given points A and B, shall be to one another in a given ratio.

Join A, B; and divide (E. 10. 6.) AB in D, so that AD is to DB in the given ratio; through D draw CEF, cutting PQ and RS, in E and F: Then

is CEF the straight line which was to be drawn. For, since PQ is (hyp.) parallel to RS, ... (E. 29. EAB=4 ABF; also (E. 15. 1.) the ADE FDB; so that the ADE, BDF are equiangular;

1.) the

AEF=▲ EFB; and the

.. (E. 4. 6.) AE: BF:: AD: DB:

But (constr.) AD is to DB in the given ratio; .. AE is to BF in the given ratio.

PROP. XXIX.

37. PROBLEM. To find a point within a given triangle, from which if three straight lines be drawn to the three angles of the triangle, it shall thereby be divided into three parts that are each to each in given ratios.

Let ABC be the given A, and let PQ, QR, RS,