# 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