1. Inversions. A geometric construction relating points O, A and B looks as follows.

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1 1. Inversions Definitions of inversion. Inversion is a kind of symmetry about a irle. It is defined as follows. he inversion of degree R 2 entered at a point maps a point to the point on the ray suh that R is the geometri mean of and, that is = R 2, or = R2. geometri onstrution relating points, and looks as follows. R C n the inversion is not defined, and is not the image of any point under the inversion. hus, the inversion is really a mapping of the plane puntured at to itself. bserve that points of the irle entered at of radius R are fixed under the inversion, points of the disk bounded by this irle are mapped to points outside the disk and vie versa. his irle is alled the irle of the inversion and the inversion is referred to as the inversion about this irle. he square of an inversion, that is an inversion omposed with itself, is the identity map. In other words, an inversion is invertible map and the map inverse to an inversion is the same inversion. he definitions of refletion about a line and inversion does not look similar. However these two transformations admit similar definitions. 1.. heorem. ny irle passing through points symmetri about a line is orthogonal to the line. For any two points non-symmetri about a line l there exists a irle passing through the points whih is not orthogonal to l. 1

2 2 C Exerise. Prove heorem 1.. his theorem allows todefine refletion about aline l asamapwhih maps a point to a point suh that any irle passing through and is orthogonal to l. 1.. heorem. ny irle passing through a point and its image under the inversion about a irle is orthogonal to. If is not the image of under the inversion about a irle, then there exists a irle passing through the points whih is not orthogonal to. u Proof. For any irle u passing through points and the degree of the enter of inversion with respet to u is. herefore 2 =, where is the segment of the line tangent to u between the enter of inversion and the point of tangeny. n the other hand, is equal to the degree R 2 of the inversion, that is to the square of the radius of irle. herefore. Hene, is a radius of. s a radius of, it is perpendiular to the tangent of. hus at the lines to u and are perpendiular to eah other. proof of the seond statement is an exerise Images of lines and irles. bviously, a line passing through the enter of an inversion is mapped by the inversion to itself. 1.C. heorem. he image under an inversion of a line l not passing through the enter of the inversion is a irle passing through and having at a tangent line parallel to l.

3 3 l Proof. Drop the perpendiular to l from. Let be its intersetion with l. Let be the image of under the inversion. ake arbitrary point l. Denote by its image under the inversion. y the definition of inversion =. herefore =. yss-test for similar triangles, is similar to. herefore =. he latter angle is right, beause l. Hene belongs to the irle with diameter. Vie versa, let us take any point of the irle with diameter. Draw a ray and denote the intersetion of this ray with l by. riangles and similar by the -test. Hene = and =. herefore, is the image of under the inversion. 1.D. Corollary. he image under an inversion of a irle passing through the enter of the inversion is a line whih is parallel to the line tangent to at. Proof. his follows from heorem 1.C, beause an inversion is inverse to itself. 1.E. heorem. he image under an inversion of a irle that does not pass through the enter of the inversion is a irle that is the image of under a homothety entered at. Proof. Let be a point of irle, and be the image of under the inversion. Denote by the seond intersetion point of the ray with. y definition of inversion, = R2, where R2 is the degree

4 4 of inversion. n the other hand, = d2, where d2 is the degree of with respet to the irle. Reall that d does not depend on the points and, this is the length of segment of a tangent line from to between and the point of tangeny. Substituting this formula to the formula for, we get = R2 d. 2 his means that is the image of under the homothety with enter and ratio R2. Hene, the image of under the inversion is the image d 2 of under this homothety. 1.F. heorem. omposition of two inversions with the same enter is a homothety entered at the same enter. he ratio of this homothety is the ratio of the degrees of the inversions. Proof. Exerise. 1.G. heorem. n inversion preserves angles between lines and irles. Proof. Let us begin with speial ases. he first ase is the angle between two lines, l and m. heir images will be irles l and m passing through, the enter of inversion. We are interested in the angle between these irles at the point whih is the image of the intersetion l m. First, let s onsider the angle at the other intersetion point, whih is the point. he angle between the irles at is by definition is the angle between their tangent lines. ut the tangent line to l (resp. m ) at is parallel to l (resp. m), so the angle between tangent lines is the same as the angle between l and m. t the other intersetion point - the point l m - the angle is the same as the angle at by symmetry. ne an also prove the same fat in a different way, as follows. Consider, first, the angle between two lines, one of whih passes through the enter of inversion. m Q P l

5 Let the lines be l and m, the enter of inversion be m. hen the image of m under the inversion is m, while the image of l is a irle passing through. he enter Q of lies on the perpendiular P dropped from to l. he angles P and P are omplementary. he angles Q and Q are equal as angles in an isoseles triangle Q. he angle Q is right as an angle between a radius Q and tangent line. herefore angles and Q are omplementary. Consequently, P =. n angle between arbitrary two lines an be represented as the sum ordiffereneofanglesbetweenthesamelinesandalinepassingthrough the enter of inversion. Consider now the angle between a line m passing through the enter of inversion and a irle whih does not pass through. See the piture: 1 1 m 5 Let be the image of under the inversion. We know from heorem 1.E that and are related by homothety entered at that sends the point 1 to. his homoethety sends the enter of one irle to the enter of the other. hus, triangles and 1 are similar, and angles and 1 are ongruent. ut is an isoseles triangle, so 1 = 1. We would like to show that angle between the line m and the irle (i.e. the angle 1 between m and the tangent line to at ) is the same as the angle between the line m and the irle (i.e. the angle between m and the tangent line to at ). ut these angles omplement the ongruent angles and 1 to right angles (sine a tangent line is perpendiualr to a radius), therefore they are ongruent. he ase of the angle between a line passing through and a irle passing through is an exerise. o omplete the proof, notie that an angle between arbitrary two irles, or an arbitrary line and an arbitrary irle, an be represented as the sum or differene of angles between the same lines or irles and a line passing through the enter of inversion. hus, the general ase redues to the speial ones above.