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The Foothills of Taniyama-Shimura. An elementary proof of Descartes' Circle Theorem. An elementary proof of the Extended (Complex) Descartes' Circle Theorem.

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The Foothills of Taniyama-Shimura. An elementary proof of Descartes' Circle Theorem. An elementary proof of the Extended (Complex) Descartes' Circle Theorem.

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A Proof of Descartes' Circle Theorem

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A Proof of Descartes' Circle Theorem. Concerning four circles, each of which touches the remaining three). Dpi{100} PQ = a b qquad QR = b c qquad RP= c a ] [ dpi{100} PS = a d qquad QS = b d qquad RS = c d. ]. Descartes' Circle Theorem states that in such a situation a,b,c,d are related by the equation. 2( frac{1}{ d} {2} frac{1}{ a} {2} frac{1}{ b} {2} frac{1}{ c} {2} ) = { left( frac{1}{d} frac{1}{a} frac{1}{b} frac{1}{c} right)} {2} ]. Theta phi psi = 2 pi. ]. Now, the fact that the angles θ/2 &...

2

A Proof of the Extended Descartes' Circle Theorem

http://www.foothills-ts.net/mother/ExtendedDescartesCircleTheorem.htm

A Proof of the Extended Descartes' Circle Theorem. Z = x iy. ]. Let p, q, r, s be the complex coordinates of the centres P, Q, R and S. Then the extended theorem says. Left { frac{p}{a} frac{q}{b} frac{r}{c} frac{s}{d} right } {2} = 2 left { frac{p {2} {a {2} frac{q {2} {b {2} { frac{r {2} {c {2} frac{s {2} { d} {2} right } . ]. And the right-hand side:. If these are to be equal for all values of w, the terms linear in w, and those involving w² must be equal separately. Equating the terms involv...Note t...

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