If points $A$, $B$, $C$ form a triangle in euclidean space and $D$ is another point in the plane of the triangle, the problem is to show that :

$\frac{AB}{DA + DB} + \frac{BC}{DB + DC} \ge \frac{AC}{DA + DC}$

I verified that it holds for more than several million positions of point $D$, including when point $D$ is inside triangle $ABC$. However, I can't seem to give a proof for it. The only inequality that resembled the one above was the Ptolemy's theorem which gives a relation between the four sides of a quadilateral made from points $A,B,C, D$ and two diagonals $AC,BD$:

$AB.DC + BC.DA \ge AC.DB$ which in other terms is $\frac{AB}{DA \cdot DB} + \frac{BC}{DB \cdot DC} \ge \frac{AC}{DA \cdot DC}$

This has multiplication in the denominator but what I am looking for is addition. Any ideas on how to go about it or perhaps some inequality results that can be applied to prove it ? Or do you see a special instance wherein the inequality doesn't hold?.

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