Trace-inequality on triangle

Question

I need to show that for $f\in C^1(T)$ on a triangle $T$ with corner points $P$ and edges $E$ with $D=\max \{|P-x|:x\in E \}$ $$ \parallel f \parallel_{L^2(E)}^2 \leq \frac{|E|}{|T|}\parallel f \parallel_{L^2(E)} (\parallel f \parallel_{L^2(T)}^2+d\parallel \nabla f \parallel_{L^2(T)}^2) $$ Now sadly I have no idea how to proof this but maybe one has to use the Poincare'- or Friedrich's inequality? Help is very much appreciated.