From the introduction: The Erdős-Mordell theorem states that if \(P\) is an interior or boundary point of the triangle \(ABC\), and \(X\), \(Y\), \(Z\) are the feet of the altitudes from \(P\) to the sides \(BC\), \(CA\), \(AB\) (produced if necessary), then
\[ PA + PB + PC \geq 2(PX+PY+PZ). \]
There is equality here if and only if the triangle is equilateral and \(P\) is the center.
Abstract: The authors show that in the Erdős-Mordell theorem, the part of the region in which the inequality holds and which lies outside the triangle, is bounded if and only if the sum of the sines of the two smaller angles is strictly greater than \(3/2\).