A theorem of Pappus's
If we draw two circumferences C and C', which are internally tangent on the point T, we can take the diameter d of the greater circumference that goes through T and draw a circumference C0 which is tangent to C and C' and has its centre on the diameter d. Then, we can draw a chain of circumferences C1, C2, C3, …(and their symmetric ones C'1, C'2, C'3,…with respect to the diameter d) disposed in such a way that every circumference is tangent to C, C' and to its preceding one.If we say that the circumference Ci has centre Oi and radius ri, the distance between Oi and the diameter d is 2iri.The figure shows this result in the case i=10 and suggests how it is possible to prove it with an appropriate circular inversion.
© matematita
The image belongs to the sections...:
Other problems (2D geometry)
Circle (2D geometry)
