$ABC$ is a triangle there are points $D, E$ and $F$ on sides $AC, AB$ and $BC$ respectively if $AF = 4$ and $BD = 12$ find the minimum value of $EC$ Does this use any specific inequality? here's my approach: since a line interior of a triange drawn from a vertice cannot exceed the neighbouring sides we can set an inequality and also all the measurements are integers

It’s simple if you allow points A&D, C&F, and B&E to coincide. Then, a variation of (assuming angle BAC is not right) the Pythagorean theorem will yield [a variation of] $(EC)^2=(AF)^2+(BD)^2$. Whatever the approach, the minimum length of EC will require the segments AF, BD, and EC each to be the entire length of their respective sides.