见
Pedoe 1970
The circle $C_0$ which is orthogonal to each of the $C_i$ is uniquely defined, unless the $C_i$ belong to a coaxal system (pencil) of circles. In the latter case the only tangent circles are two point-circles, in the case when the coaxal system is of the intersecting type. The circle $C_0$ plays a very special rôle with regard to the $C_i$ since inversion (transformation by reciprocal radii) in $C_0$ maps each $C_i$ onto itself, and maps a tangent circle $C$ onto a tangent circle $C^{\prime}$. When there are 8 tangent circles (which may be called the general case) these can be split into 4 pairs. We shall call the circles in a pair conjugate circles. (For all this, proved algebraically, see Pedoe [3]). If we wish to specialize the $C_i$ so that there are only 7 tangent circles, the specialization must aim at making a pair of conjugate circles identical, since if two tangent circles which are not conjugate become identical the conjugates also become identical, and the number of tangent circles would reduce to 6, at most.
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We therefore specialize the $C_i$ so that a conjugate pair $C$ and $C^{\prime}$ become the same circle $D$, say. This means that inversion in $C_0$ maps the tangent circle $D$ onto itself. If this is the case, $D$ must be orthogonal to $C_0$. We therefore find ourselves with three circles $C_i$, a circle $C_0$ orthogonal to the $C_i$, and a circle $D$ which touches the $C_i$ and is also orthogonal to $C_0$. We show that this means that two of the $C_i$ touch each other.
Invert with respect to a centre of inversion on $C_0$. We obtain three circles $C_i^{\prime}$, with diameters which lie along the line $C_0^{\prime}$.
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These three circles are touched by a circle $D^{\prime}$ whose diameter also lies along $C_0^{\prime}$. If two circles with diameters along the same line touch at a point not on this line, they have the same centre, and must therefore coincide. If the circles are distinct contact can only take place at an endpoint of a diameter. Since $D^{\prime}$ has only 2 points of intersection with the line $C_0^{\prime}$, and has to touch each of $C_1^{\prime}, C_2^{\prime}$ and $C_3^{\prime}$ at a point on $C_0^{\prime}$, the three points of contact cannot be distinct. Hence at least two of the circles $C_i^{\prime}$ intersect $C_0^{\prime}$ at the same point. That is, at least two of the circles $C_i^{\prime}$ touch each other. But if at least two of the circles $C_i$ touch each other, the number of circles tangent to the three $C_i$ is readily seen to be 6, at most.
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Post time 2023-9-17 21:09
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