Positive operator on a complex Hilbert space is symmetric?
We have $$ \mathbb{R}\ni\langle A(x+y),x+y\rangle=\langle Ax,x\rangle+\langle Ay,y\rangle+\langle Ax,y\rangle+\langle Ay,x\rangle. $$ Thus $\langle Ax,y\rangle+\langle Ay,x\rangle=\langle x,Ay\rangle+\langle y,Ax\rangle$.
Moreover, $$ \mathbb{R}\ni \langle A(x-iy),x-iy\rangle=\langle Ax,x\rangle+\langle Ay,y\rangle+i\langle Ax,y\rangle-i\langle Ay,x\rangle, $$ which implies $\langle Ax,y\rangle-\langle Ay,x\rangle=\langle x,Ay\rangle-\langle y,Ax\rangle$.
If you add up these to equalities, you arrive at $\langle Ax,y\rangle=\langle x,Ay\rangle$.