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describe a circle which shall touch both PB and TD, and which shall also touch the circle GH.

Find (E. 1. 3.) the centre L of the circle GH; from D draw (E. 11. 1.) DX 1 to TD and make it equal to the semi-diameter of HG; through X draw (E. 31. 1.) XW parallel to TD; also, as in S. 90. 3, draw RK, equi-distant from PB and TD; describe (S. 89. 3.) a circle which shall have its centre in RK, which shall pass through L, and touch WX; let K be the centre of the circle, so described, and let it touch WX in X; join K, M and K, L; and let KM and KL cut TD, and the circumference of GH, F and G, respectively: Then, since (E. 18. 3.) the KMW is a right 4,

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and that (constr.) WX is parallel to TD, .. the ▲ KFT is also a right ; and, because (constr.) MD is a□, (E. 34. 1.) MFXD; but (constr.) XD=LG; ... MFLG; and (constr. and E. 15. def. 1.) KM=KL; .. KF=KG, and a circle described from the centre K, at the distance KF, will (E. 16. 3. cor.) touch TD in F, will pass through G, and (S. 6. 3.) will touch the circle HG in G.

PROP. XCII.

119. PROBLEM. To describe a circle which shall touch both a given circle, and a given straight line, and which shall, also, pass, first, through a given point without the given circle; and, secondly, through a given point within the circle.

Let BCH be the given circle, PQ the given

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straight line, and first let the given point A be without the circle: It is required to describe a

circle which shall pass through A, and which shall touch both PQ and the circle BCH.

Find (E. 1. 3.) the centre K of the circle BCH, and draw (E. 12. 1.) the diameter BDKC to PQ; join C, A,* and produce CA to E (S. 67. 3. cor.) or divide it (S. 71. 3. cor. 3.) so that ECX CABC X CD; describe (S. 88. 3.) a circle AEF, which shall pass through A and E, and touch PQ: It shall also touch the circle BCH.

For, let the circle AEF touch PQ in F; find its centre G; and draw the diameter FGL, which (constr. E. 18. 3. and E. 28. 1.) is parallel to BC; join, B, F, and C, F; and let CF cut the circumference of BCH in H; join, also, B, H and F, H and K, H; and let KH meet FL in G; upon BF as a diameter, describe the circle BDHF, which, because the BDF, BHF (constr. and E. 31. 3.) are right, will pass (S. 29. 1. cor. 2.) through D and H; .. (E. 36. 3. cor.) BC × CD=FC × CH; but (constr.) BC X CD= ECX CA; .. FC X CH-ECX CA, and, .., the point H is in the circumference of the circle AEF; otherwise (E.

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* If AC2 > BC × CD, then CA must be divided into two parts, so that the rectangle contained by AC and the segment toward C shall be equal to BC x CD. Also, in this application of S. 67. 3, CA must first be produced, so that the rectangle contained by CA and the part produced, shall be of the given magnitude; and then from the whole line, CE must be cut off equal to the part produced.

36. 3. cor.) the greater of two rectangles would be equal to the less. The point is, .., common to both the circles AEF, BCH.

And, since (constr.) FL is parallel to BC, .. (E. 29. 1.) the GFH = HCB; but, since (constr. and E. 31. 3.) the ▲ BHC is a right 4, the HCB+ CBH=(E. 32. 1.) a right ▲ ; .. the ▲ GFH+ 2 KBH = a right ≤; that is (E. 15. def. 1. and E. 5. 1.) the GFH + ▲ KHB = a right ; and (constr. and E. 31. 3.) the BHF is a right; .. (E. 13. 1.) the KHB + 2 GHF= a right;., the GFHGHF, and (E. 6. 1.) GF = GH: But G is in the diameter of the circle AEF; .. (E. 7. 3.) G is the centre of the circle AEF, which.. (S. 6. 3.) touches the circle BCH in H.

And, in a similar manner, the problem may be solved, when it admits of a solution, if the given point be within the given circle: It is manifest, however, that, in this latter case, the given straight line which is to be touched cannot lie wholly without the given circle.

PROP. XCIII.

120. PROBLEM. In a straight line of indefinite length, but given in position, which cuts a given circle, to find a point, from which if a straight line be drawn to touch the circle, it shall be equal to a given finite straight line.

Let LM be a given finite straight line, PAB a

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straight line given in position, but indefinite in length, cutting the given circle ABC in A and B: It is required to find a point in PB, from which, if a tangent be drawn to the circle ABC, it shall be equal to L.

Produce (S. 73. 3.) AB to D so that AD × DB: LM'; and from the centre D, at a distance LM, describe a circle cutting the circumference of the circle of ABC in C; draw DC; .. DC=LM; →. but (constr.) AD × DB=LM2; .. AD × DB X

=DC2; .. (E 37. 3.) DC touches the circle ABC, in C; and (constr.) it is equal to LM, and is drawn from a point D in the given indefinite straight line PAB.

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