Question 48
The vertices $X$, $Y$, $Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$. Prove that the incenter of triangle $ABC$ lies inside triangle $XYZ$.
The vertices $X$, $Y$, $Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC$, $CA$, $AB$ of a triangle $ABC$. Prove that if the incenter of triangle $ABC$ lies outside triangle $XYZ$, then one of the angles of triangle $ABC$ is greater than $120^{\circ}$.
The vertices $X$, $Y$, $Z$ of an equilateral triangle $XYZ$ lie respectively on the sides $BC$, $CA$, $AB$ of a triangle $ABC$. Prove that if the incenter of triangle $ABC$ lies outside triangle $XYZ$, then one of the angles of triangle $ABC$ is greater than $120^{\circ}$.