By B.H. Gross, B. Mazur
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Additional info for Arithmetic of elliptic curves with complex multiplication
5, the quadrilateral M1 M1 M2 M2 can be inscribed in a circle and thus it holds (see Fig. 30) M1 M1 M2 = M2 M2 A. Even to the limit, this property of inscribability still holds, hence M1 → M 1 ⇔ M2 → M2 . In this case, we deduce that XM 1 M2 = M1 M2 Y . 6 Inversion 43 Fig. 31 Tangent to a curve (Sect. 8) Therefore, if the tangents are intersecting at a point P then the triangle PM 1 M2 has to be isosceles. In general, these tangents have to be symmetrical with respect to the axis of symmetry determined by the perpendicular bisector of the line segment M1 M 2 .
85) It follows that AK is of constant length, and consequently the point T is moving on a fixed circle with center P and radius r= R2 − AK 2 2 2. Simultaneously, from the right triangle KBM, with B = 90°, we derive KT · KM = KB2 = R 2 . 86) The assertion follows. , USA, 2001 . Problem Let FBD be an acute triangle. Let EFD, ABF, and CDB be isosceles triangles exterior to FBD with EF = ED, AF = AB, and CB = CD, and such that FED = 2BFD, BAF = 2FBD, DCB = 2FDB. 52 3 Fundamentals on Geometric Transformations Fig.
17). (i) The center of homothety is homothetic with respect to itself. (ii) The points M, M are called homologous or corresponding points. 2. , S is the geometrical locus of the homothetic points of S) (see Figs. 19). (i) If r = 1 then S ≡ S . (ii) If M is a point homothetic to M with respect to center O and ratio r, then M is homothetic to M with respect to the point O and ratio 1/r. 5 Homothety 31 Fig. 18 Homothety (Sect. 5) Fig. 19 Homothety (Sect. 5) 3. A characteristic criterion of homothety.
Arithmetic of elliptic curves with complex multiplication by B.H. Gross, B. Mazur