Question

If $x,y,z$ are respectively perpendicular from the circumcenter on the sides of the $△ABC$, the value of $\frac{a}{x}+\frac{b}{y}+\frac{c}{z}-\frac{abc}{4xyz}$ is

Answer 1

From the figure, $\tan A=\frac{a}{2x}$

Similarly, $\tan B=\frac{b}{2y}$ and $\tan C=\frac{c}{2z}$

$\ln △ABC$

$\tan A+\tan B+\tan C=\tan A\tan B\tan C$

$\Rightarrow \frac{a}{x}+\frac{b}{y}+\frac{c}{z}-\frac{abc}{4xyz}=0$