>>14257758First I remembered a vague image of quora answer about a similar question, where an inequality was involved. One insight there was to notice that the expression is homogeneous in (a,b,c), that multiplying them all by a scalar doesn't change the value.
Then I also noticed that the second equation is not homogeneous. If you take in lambda as a parameter and replace (a,b,c) with (la,lb,lc) you'll quickly see that either the expression is zero or it can take every possible value. Since the latter is unlikely, it must be zero.
I rescaled to have D=1, so the expression becomes a^2/(1-a) + ....
At this point I noticed that a + a^2/(1-a) gives a/(1-a). Looking back at it, it was reasonable to approach it by long division, i.e. dividing the polynomial a^2 by the polynomial 1-a, getting the remainder a, but this is not how I got to it. I think I just tried to add a to it to see what I get, and I like it. Then I scaled back because the normalization was completely unnecessary for the proof.
Hope this helps.
