By Barry Spain, W. J. Langford, E. A. Maxwell and I. N. Sneddon (Auth.)

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**Thirteen Books of Euclid's Elements. Books X-XIII**

After learning either classics and arithmetic on the college of Cambridge, Sir Thomas Little Heath (1861-1940) used his time clear of his task as a civil servant to post many works just about old arithmetic, either well known and educational. First released in 1926 because the moment variation of a 1908 unique, this booklet comprises the 3rd and ultimate quantity of his three-volume English translation of the 13 books of Euclid's components, masking Books Ten to 13.

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60 ANALYTICAL GEOMETRY The straight line AXA2 is a tangent if both points Ρλ and 1\ coincide with Lx (Fig. 22). In this case, Joachimsthal's quadratic equation FIG. 22 has equal roots and so Thus the locus of the point A2 as Ax is held fixed is given by SXS-T? = 0. This quadratic equation then represents the pair of tangents AXLA and AXL2 to the circle. 61 CIRCLE EXAMPLES 43. Show that the pair of tangents from (—1, 3) to the circle x2+y2 = 5 are mutually perpendicular. 44. y2-j-8A-+6y+21 = o and calculate the angle between them.

25. Find the condition that y = mx+c intersects the circle x2~\-y2 = r2 in two distinct real points. 30. JoachimsthaPs equation Let the two points Ax(xl9 yj and A2(x2, y2) intersect the circle S ΞΞ x*+y*+2gx+2fy+c = 0 at Λ and P2 (Fig. 19). The coordinates of the point P which divides AXA2 in the ratio λ2/λ1 (section 4) are ( ( λ Λ + λ ^ / ί λ ^ λ2), ( λ Λ + ^)Κλχ+ λ2) ). 53 CIRCLE If this point P lies on the circle, we have \ λ 1+ λ 2 / \ λ 1+ λ 2 / \ λ χ +λ 2 y ^ λ 1 +λ 2 / + On multiplication by (Λ+λ,,)2, this equation simplifies to FIG.

Obtain the locus of a point which moves so that its distance from the straight line 2x—5y— 1 = 0 is five times its distance from the point (1, 2). 4. Show that the points ( - 1 , - 4 ) , (1, - 3 ) , ( - 2 , - 2 ) and ( - 1 , - 1 ) are concyclic. 5. Obtain the equations of the two lines through the point of intersection of x+6y—7 = 0 and 3x—2y+2 = 0 perpendicular to them. 6. Obtain the equations of the straight lines through (—2, 1) which make an angle of 45° with the line 3y—2x = 2. 7. Find the coordinates of the circumcentre of the triangle formed by the straight lines 3*-j>-5 = 0, x+2j>-4 = 0 and 5jt+3j> + l = 0.