An extension of Carnot’s theorem

We consider the following problem and the special cases of it as the applications. Let I be an arbitrary point in plane of triangle ABC such that I(\alpha,\beta,\gamma) in barycentric coordinates, let d_a,d_b,d_c be signed distance from I to side BC,CA,AB of triangle ABC. Then xd_a+yd_b+zd_c=\frac{x\alpha bc+y\beta ca+z\gamma ab}{2R(\alpha+\beta+\gamma)} with circumradii R.

extendcarnot.pdf

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