Geometry III/IVCircles in Euclidean geometrySome facts and proofsaaaaaFacts ...Fact 1.For any three non-collinear points on the Euclidean planeE2there exists aunique circle passing through these three points.Fact 1’.Every triangle has a unique circumscribed circle.Fact 2.Three perpendicular bisectors of a triangle do intersect in one point.Fact 3.For a circlecand two pointsX, Yoncthere exists a unique line or circleorthogonal tocand passing throughXandY.Fact 4.An inversion with respect to a circlecpreserves lines and circles orthogonaltoc.Fact 5.Lines and circles not orthogonal to a circlecare not preserved by inversionwith respect toc.Fact 6.Letcbe a circle and letA,Bbe two points not inc. Then there exists aunique line or circle orthogonal tocand passing throughAandB.Fact 7.Letcbe a circle andc′be a line or circlec⊥c′. LetAbe any point not inc∩c′. Then there exists a unique line or circlec1passing throughAand orthogonalto bothcandc′.Fact 8.Stereographic projection coincides with the restriction to the sphere of asuitable inversion.Fact 9.LetABbe a diameter of a circlecandCany other point. Then∠ACB=π/2 if and only ifC∈c.aaaaaaaaaaaaaa... and ProofsFact 1.For any three non-collinear points on the Euclidean planeE2there exists aunique circle passing through these three points.Proof.LetA, B, Cbe three non-collinear points. LetMbe a midpoint of the seg-mentABand letNbe a midpoint ofBC. Letl1be a line throughMorthogonaltoABand letl2be a line throughNorthogonal toBC. SinceA, BandCare notcollinear,l1is not parallel tol2. Denote byOtheir intersection pointO=l1∩l2(see Fig.1.a).