\frac yzBP+\frac zyBN=\left(\frac yz+\frac zy\right)\left(\frac {BP+BN} 2\right)-r\left(\frac {yx}z-\frac{zx}y\right)\geq (BP+BN)-r\left(\frac {yx}z-\frac{zx}y\right)