By I. M. Yaglom, Henry Booker, D. Allan Bromley, Nicholas DeClaris

ISBN-10: 1483256634

ISBN-13: 9781483256634

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Hence it follows that the angle δ is equal to the angle δ0 between the lines which intersect at O and pass through the points #J and z\ ; that is δ = ar SV ? (= y\ ar S(^2 - *o) - g(*i - *o)) (8) r / 'h'h FIG. ±—^ %! - Z0 the ratio of the three points (three complex numbers) z2 ,zlyz0. Thus, the angle δ between the lines which intersect at a point ζ$ and pass through the points zx and z2 is equal to the argument of the ratio V(z2 , ζλ , z0) of the points z2 , ζλ , ζ0 . We note that δ is the angle through which the ray z0z1 of the first line must be rotated counterclockwise to be brought into line with the ray z0z2 of the second line.

And w2 lie on one cirWe use the fact that the points zliz2iw11 cle $ 2 , the points z2, z3, w2, and w3 on the circle S3 , the points z3 , #4 , w3 , and w± on the circle £ 4 , and the points # 4 , ^ , w4 , 35 §8. Applications and Examples FIG. 7 a n d w1 on t h e circle S± . 8) W ( * 2 , w 3 , ar3 , w 2 ) · W{z± ,wlyzly \ Z~ — Z9 ' Z,—Z± w4) ) \ HI, — Wo Vo — W,) is real. T h e r e f o r e , since t h e cross-ratio W{zx , z3 , z2 , zá) is real, it follows that t h e cross-ratio W(w1 , w3 , w2 , w 4 ) is real, w h i c h proves the theorem.

22 We know already that Equation 14 denotes a line if {and only if) A=0 (15) 22 Equation 14 can also be deduced from the fact that by the translation given by Equation 3 any circle S can be taken into the circle zz = r2 with center at origin O (r is the radius of the circle). Hence we obtain the equation of S, (z + q)(z + q) — r2 = 0 or azz + bz + bz + c = 0, where a = 1 and c = qq — r2 are real, and where b = q. From this equation it is obvious that the square of the radius r of the circle »S is equal to (bh — ac)/a2 and its center is the point —b/a.

### Complex Numbers in Geometry by I. M. Yaglom, Henry Booker, D. Allan Bromley, Nicholas DeClaris

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