I've been reading a book, and in the geometry section, it gives what seem to me unnecessary and complicated proofs for simple facts.
For example:
$D,E,F$ are the feet of the altitudes from $A,B,C$ of $\Delta ABC$. Prove that $AD+BE+CF<AB+BC+AC$
Proof:
In $\Delta ABD$, $AD<AB$. In $\Delta ADC$, $AD<AC$. Adding, we get $2AD<AB+AC$.
Similarly, $2BE<AB+BC$ and $2CF<AC+BC$. Adding, we get $2(AD+BE+CF)<2(AB+BC+AC)$. Dividing by 2, we get the required result.
This seems to me unnecessary. Why not just use $AD<AB,BE<BC,CF<AC$ and add to get the required result?
Another one, in proving that $AB-AC<BC$, takes a similarly long winded approach, constructing $D$ on $AC$ such that $AB=AD$ and then going from there. I just used the triangle inequality, $AB<AC+BC$, and subtracting $AC$ from both sides.
Is there something wrong with my approach? Why does this book, and a lot of other books on the market, give such proofs?