Let CAB be the given 2 ; let AK, AL, be the two

straight lines given in position; and let AL be to AK in the given ratio: It is required, first, to find, within the CAB, the locus of all the points, from which, if straight lines be drawn to AC and AB, parallel to AL and AK, respectively, they shall be to one another as AL to AK.

Through K and L draw (E. 31. 1.) KM, and LM, parallel to AB and AC, respectively, and meeting in M; draw AM, and produce it, indefinitely, toward X; AX is the locus which was to be found.

For take any point P in AX, and from P draw PQ parallel to AL, and PR parallel to AK: And, since (constr. and E. 29. 1.) the APR, KAM are equiangular, as are, likewise, the APQ, MAL.

.. (E. 4. 6.) PR: AK:: AP: AM:: PQ: AL .. (E. 11.-5.) PR: AK::PQ: AL

.. (E. 16. 5.) PR: PQ:: AK : AL.

Secondly, let B and C be two given points in AB and AC: It is required to find the locus of

all the points, from which if straight lines be drawn parallel to AK and AL, they shall cut off from CA and BA two segments, which are always to one another in the same ratio as the given finite straight lines AK and AL,

From CA cut off CE=AK, and from BA cut off BF=AL; from C and B, draw (E. 31. 1.) CD parallel to AL, and BD parallel to AK, and let CD and BD meet in D; likewise from E and F draw EG parallel to AL or CD, and FG parallel to AK or BD, and let EG and FG meet in G: Through D and G draw the straight line DGX: Then is DGX the locus which, in this case, was to be found.

For take any point in it, as P, and draw PH parallel to DC, and PI parallel to DB: Then it is manifest from the demonstration of E. 10. 6. that

X

HC: EC:: PD: GD:: IB: FB;

.. (E. 16. 5.) HC: IB:: EC: FB:

That is (constr.) HC is to IB in the given ratio: And it is easily shewn, ex absurdo, that no point which is out of the locus so determined, has the property described in the proposition.

13. COR. The intersection of the one locus with the other, determines a point, from which if two straight lines be drawn to AB and AC, in the given directions, they shall be to one another in the same given ratio as the segments are, which they cut off from CA and BA.

PROP. VIII.

14. THEOREM. If a circle be touched, in the same point, both externally and internally, by two other circles, and through the point of contact two straight lines be drawn, the parts of them intercepted between the circumference of the given circle, and that of the circle which touches it internally, shall have to one another the same ratio as the parts which are chords of the other circle.

Let the given circle ABC be touched in the same point A, internally by the circle DAE, and externally by the circle FAG; and through A let there be drawn any two straight lines, BAG, CAF,

each cutting the three circles ABC, DAE, FAG: Then BD:CE:: AG: AF.

For, draw BC, DE, and FG; and through A draw (E. 17. 1.) HAL touching the circle BAC, in A, and ... touching the two circles DAE, FAG: And since (E. 15. 1.) the DAH LAG, and that (E. 32. 3.) the DAH=4DEA, and the <LAG= AFG, .. the DEF EFG, and

(E. 27. 1.) FG is parallel to DE: Also, since (E. 32. 3.) the DAH or BAH, is equal to each of the DEA, BCA, they are equal to one another, and .. (E. 28. 1.) BC is parallel to DE; .. (E. 2. 6.) BD:CE:: AG: AF.

PROP. IX.

15. PROBLEM. From the centre of a given circle, to draw a straight line to meet a given tangent to the circle, so that the segment of the line between

the circle and the tangent shall be any required part of the tangent.

Let ABC be a given circle, of which K is the

centre, and let BD touch the circle in B: It is required to draw a straight line from K to BD, so that the segment of it, between the circle and BD shall be any required part of the segment BD.

Draw KB; divide (S. 49. 1.) KB into a number of equal parts, equal to the number of times which the segment of BD is to contain the segment of the straight line to be drawn from K to BD; and from BD cut off BF equal to one of them; from F draw (E. 17. 3.) FC touching the circle ABC in C; through C draw KCD: Then shall CD be the required part of BD.

For (constr. and S. 26. 1.) the two KBD,