However it makes the essay more interesting and we

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the syllabus, it should be possible to get full marks without including it. However, it makes the essay more interesting and we would certainly get credit for it.] 10
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Theorem 4.5 (Steiner’s porism) . Suppose C and C are non intersecting circles with C inside C . If C 1 , C 2 , . . . form a sequence of circles arranged in clockwise order with C 1 tangent to C and C and C j +1 tangent to C and C and C j [ j 1] we say that they form a chain of circles. If C n = C 1 we say that the chain is closed. Steiner’s porism states that if one choice of C 1 gives a closed chain then all choices do. [This should be illustrated by a diagram. Use diagrams freely throughout your essay. Remember, a picture is worth a thousand words.] Sketch proof. We can find a M¨obius transform which takes C and C to concentric circles. The transform takes chains of circles to chains. Since Steiner’s porism is trivial for concentric circle we are done. [This really is a sketch. It is not obvious without some further argument that there is a M¨obius transform which takes C and C to concentric circles. We have not shown explicitly that tangent circles are taken to tangent circles.] Just as a simple translation allows us to move any point in R n to the origin so M¨obius maps allow us to move any three points in C to any other. [Note that we are trying to show that M¨obius maps are useful. A skeptical reader would not be fully convinced without the inclusion of material from later courses.] Here is a particular example. Lemma 4.6. (i) If w negationslash = , the M¨obius map z mapsto→ 1 / ( z w ) takes w to . (ii) If w negationslash = the M¨obius map z mapsto→ z w takes w to 0 and fixes . (iii) If w negationslash = 0 , the M¨obius map z mapsto→ z/w takes w to 1 and fixes 0 and . (iv) If w 1 , w 2 , w 3 are distinct then there exists a M¨obius map T with Tw 1 = 0 , Tw 2 = 1 and Tw 3 = Using Lemma 4.6 it can be shown that the following general result holds. [We are not obliged to prove everything, but it helps the reader if we make it clear when we are not proving something.] Theorem 4.7. If w 1 , w 2 , w 3 are distinct and z 1 , z 2 , z 3 are distinct then there exists a M¨obius map T with Tw j = z j The map T of Theorem 4.7 is unique. To show this we introduce the idea of the cross ratio [ z 1 , z 2 , z 3 , z 4 ]. [Generally speaking it is bad exposition to introduce a definition that is used only once, but this is an exam and we wish to display our knowledge.] 11
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Definition 4.8. We set [ z 1 , z 2 , z 3 , z 4 ] = ( z 1 z 2 )( z 3 z 4 ) ( z 1 z 4 )( z 3 z 2 ) treating as as in the definition of the M¨obius map. [This is more of sketch of a definition than a definition] Lemma 4.9. If T is a M¨obius map [ Tz 1 , Tz 2 , Tz 3 , Tz 4 ] = [ z 1 , z 2 , z 3 , z 4 ] . Sketch proof. The result is true for elementary M¨obius maps by direct cal- culation and so true for all M¨obius maps by Lemma 4.1 (iii).
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