In geometry, Barrow's inequality states the following: Let P be a point inside the triangle ABC; U, V, and W be the points where the angle bisectors of BPC, CPA, and APB intersect the sides BC, CA, AB, respectively. Then
\( PA+PB+PC\geq 2(PU+PV+PW).\, \)
Barrow's inequality strengthens the Erdős–Mordell inequality, which has a similar form with PU, PV, and PW replaced by the three distances of P from the triangle's sides. It is named after David Francis Barrow.
Euler's theorem in geometry
Hojoo Lee: Topics in Inequalities - Theorems and Techniques
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