page_title: Descartes' theorem
section_title: Special cases
hierarchical_section_title: Descartes' theorem / Special cases
caption_attribution_description: English: One of the circles is replaced by a straight line of zero curvature. Descartes' theorem still applies.
context_page_description: In geometry, Descartes' theorem states that for every four kissing, or mutually tangent, circles, the radii of the circles satisfy a certain quadratic equation. By solving this equation, one can construct a fourth circle tangent to three given, mutually tangent circles. The theorem is named after René Descartes, who stated it in 1643.
context_section_description: If one of the three circles is replaced by a straight line, then one kᵢ, say k₃, is zero and drops out of equation (1). Equation (2) then becomes much simpler:
  If two circles are replaced by lines, the tangency between the two replaced circles becomes a parallelism between their two replacement lines. For all four curves to remain mutually tangent, the other two circles must be congruent. In this case, with k₂ = k₃ = 0, equation (2) is reduced to the trivial
  It is not possible to replace three circles by lines, as it is not possible for three lines and one circle to be mutually tangent.
  Descartes' theorem does not apply when all four circles are tangent to each other at the same point.
  Another special case is when the kᵢ are squares,
  Euler showed that this is equivalent to the simultaneous triplet of Pythagorean triples,
  and can be given a parametric solution.  When the minus sign of a curvature is chosen,
  this can be solved as,
  where
  parametric solutions of which are well-known.
caption: One of the circles is replaced by a straight line of zero curvature. Descartes' theorem still applies.
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