Question 54
Let $ABC$ be a triangle. The moving points $M$ on the half line $BC$, $N$ on the half line $CA$, and $P$ on the half line $AB$, start simultaneously from vertices $B$, $C$, and $A$, respectively, and move with constant speeds $v_1, v_2, v_3 \gt 0$, expressed using the same unit.
a) Knowing that there are three distinct moments in which the triangle $MNP$ is equilateral, prove that the triangle $ABC$ is also equilateral and $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which the triangle $MNP$ is equilateral, then the triangle $ABC$ is also equilateral.
a) Knowing that there are three distinct moments in which the triangle $MNP$ is equilateral, prove that the triangle $ABC$ is also equilateral and $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which the triangle $MNP$ is equilateral, then the triangle $ABC$ is also equilateral.