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analytic geometry angle bisector angles subtended axiom axis chord circumcenter circumcircle collinear common Compute concurrent lines concyclic congruent contains convex polygon cosets crystallographic groups cyclic defined definition diagonal diameter draw edges elementary angles elements elliptic geometry equal euclidean geometry Exercise exists fixed point form a pencil formula Gauss plane glide reflection group G halfplane hence homothety hyperbolic geometry hyperbolic lines identity implies inequality inscribed integer interior inversion involution isogonal isomorphic isosceles isotropy group lattice length line g M-circles midpoints motion multiplication normal subgroup notation obtain one-to-one oriented angles orthic triangle pairs parallel lines parallelogram perimeter perpendicular bisector point of intersection point reflections points of contact problem product of reflections proof Prove quadrilateral ratio real numbers rectangle right triangle rotation segments sides similitude square tangent theorem three lines three points three reflections translation triangle A(A,B,C unique vector vertex vertices