Let $ABC$ be an acute triangle with $AX, BY$ and $CZ$ as its altitudes. $\bullet$ Line $\ell_A$, which is parallel to $YZ$, intersects $CA$ at $A_1$ between $C$ and $A$, and intersects $AB$ at $A_2$ between $A$ and $B$. $\bullet$ Line $\ell_B$, which is parallel to $ZX$, intersects $AB$ at $B_1$ between $A$ and $B$, and intersects $BC$ at $B_2$ between $B$ and $C$. $\bullet$ Line $\ell_C$, which is parallel to $XY$ , intersects $BC$ at $C_1$ between $B$ and $C$, and intersects $CA$ at $C_2$ between $C$ and $A$. Suppose that the perimeters of the triangles $\vartriangle AA_1A_2$, $\vartriangle BB_1B_2$ and $\vartriangle CC_1C_2$ are equal to $CA+AB,AB +BC$ and $BC +CA$, respectively. Prove that $\ell_A, \ell_B$ and $\ell_C$ are concurrent.