Classical Theorems
Click the figure below to see the complete classical theorem.
Sunday, May 8, 2011
Online Geometry: Carnot's Theorem in an Acute Triangle
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Classical Theorems
Click the figure below to see the complete classical theorem.
r=2*area/(a+b+c)=(ad+be+cf)/(a+b+c). Plugging into the proposed equation and multiplying by (a+b+c) gives d(b+c)+e(a+c)+f(a+b)=R(a+b+c). From Ptolemy's theorem applied to each of the quadrilaterals formed from the perpendicular bisectors and sides of the triangles, we get Ra/2=ec/2+fb/2, or Ra=ec+bf. Summing cyclically gives R(a+b+c)=ec+bf+fa+cd+db+ae=d(b+c)+e(a+c)+f(a+b).
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