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

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**Example text**

FIG. 9a, b, c, d. 42 II. , n, with points "i > u2 > u3 >···> un on them, lying on one line or circle. s . , un of all possible combinations of n — 1 of the given lines lie on one circle, for which it is sufficient to ensure that any four of these points, for example the points ux , u2 , u3 , and ux , lie on one circle. ··» Sn of all possible combinations of n — 1 of our lines meet in one point, and for this it is sufficient to ensure that any three of these circles, say S± , S2 , and S3 , meet in one point.

However, we do not limit ourselves to these, 57 §8. Applications and Examples and we mention here some more theorems which characterize these points. First, it is easy to see that m = ~ ax + a2 + az + a 4 4 = (ax + a2)¡2 + (tf3 + 2 2 a^jl 2 Hence it follows that the point m is the common midpoint of the segments m12mM , m1¿m23 , and ra13w24 , which join the midpoints m12 and ra34 of the opposite sides axa2 and a3a4 of the quadrangle, the midpoints mu and ra23 of the opposite sides a±aA and a2a3 , and the midpoints mu and m24 of the diagonals axa^ and ¿^¿ζ ; the three segments joining the midpoints of opposite sides and the midpoints of the diagonals of the quadrangle αλα2α^α^ meet in one point m, the centroid of the quadrangle, and are bisected there; see Figure 16.

Cauchy (1789-1857). This interpretation arises from the fact that the point of a plane with rectangular cartesian coordinates x and y or polar coordinates r and φ corresponds to the complex number (see Figure 1): z = x + iy = r(cos φ -\- i sin ψ) Here, obviously, real numbers z = x + 0 · i = r(cos 0 + i sin 0) correspond to points of the x axis, the real axis o\ numbers of modulus r = 1 correspond to points of the circle S with center at O and radius 1, the unit circle. Opposite complex numbers z = x + iy and — z = —x — iy correspond to points symmetrical about the point O (the number 0 corresponds to the origin isa T h e contents of Sections 7, 8, 13 and 14 have many points of contact with the books of R.