By Kollar J., Lazarsfeld R., Morrison D. (eds.)
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Additional resources for Algebraic Geometry Santa Cruz 1995, Part 1
J 16. An inversion is permutable with a symmetry with respect to a line only if its center is on the line. The product is then an involution. ] 25. III. DEFINITION AND INTERPRETATION 35 ANHARMONIC RATIO 25. Definition and interpretation. R. FIG. 16 as a function of geometric elements, place arbitrary axes a 13 , au, a 23 , a 24 on the lines ZIZs, Z IZ4' Z2Z3, ZZZ4' which mayor may not be distinct, and designate by Z IZ3, for example, the algebraic value calculated on a l3 of the segment having Zl for initial point and Z3 for terminal point.
Show that the inversions are permutable only if their powers are opposite and that the homothety is then a symmetry. 15. Two inversions having distinct centers are permutable only if the square of the distance between the centers is the sum of the powers of inversion, or, in other words, only if the director circles or circles of double points are orthogonal. Their product is then an involutoric transformation (a Mobius involution, see article 99). J 16. An inversion is permutable with a symmetry with respect to a line only if its center is on the line.
Centroid of a triangle. If a, b, c are the affixes of the vertices A, B, C of a triangle ABC, which may be a degenerate triangle, then the affix g of the centroid G of y the triangle is I g = "3 (a B + b + c). If A' is the midpoint of segment BC, we have C a' o~----------x and since = l(b + c) 2 AG GA' =2, FIG. 25 we have (35) g= a +2a' I I + 2 = 3" (a + b + c). 39. Algebraic value of the area of a triangle. If the angle (xy) has the value 1T/2 and if in tracing the perimeter of a triangle one meets the vertices A, B, C of affixes a, b, c in this order, then the algebraic value ABC of the area of the triangle is + ABC = ~.