KöMaL 2010 A515

There is given a triangle ABC . For every 0<$t$<1 let $U$($t$) and $V$($t$) be the points which divide the line segments $AB$ and $BC$ in the ratio $t$:(1-$t$), respectively. Prove that there is a parabola which is tangent to all lines $U$($t$)$V$($t$).