ABC

BC

CA

AB

D

E

F

AD

BE

CF

X

Y

Z

\frac{1}{\frac{AX}{XD}+\frac{1}{4}}+\frac{1}{\frac{BY}{YE}+\frac{1}{4}}+\frac{1}{\frac{CZ}{ZF}+\frac{1}{4}}=4.

AE=AF=x,~BD=BF=y,~CD=CE=z.

AB^{2}\cdot DC+AC^{2}\cdot BD-AD^{2}\cdot BC=BC\cdot DC\cdot BD~\Rightarrow

\Rightarrow~AD^{2}=\frac{AB^{2}\cdot DC+AC^{2}\cdot BD-BC\cdot DC\cdot BD}{BC}=

=\frac{(x+y)^{2}z+(x+z)^{2}y}{y+z}-yz=x^{2}+\frac{4xyz}{y+z}.

AX\cdot AD=AF^{2}=x^{2}~\Rightarrow~AX=\frac{x^{2}}{AD}~\Rightarrow

\Rightarrow~XD=AD-AX=AD-\frac{x^{2}}{AD}=\frac{AD^{2}-x^{2}}{AD}~\Rightarrow

\Rightarrow~\frac{AX}{XD}=\frac{\frac{x^{2}}{AD}}{\frac{AD^{2}-x^{2}}{AD}}=\frac{x^{2}}{AD^{2}-x^{2}}=\frac{x^{2}}{x^{2}+\frac{4xyz}{y+z}-x^{2}}=\frac{x(y+z)}{4yz}.

\frac{BY}{YE}=\frac{y(z+x)}{4zx},~\frac{CZ}{ZF}=\frac{z(x+y)}{4xy}.

\frac{1}{\frac{AX}{XD}+\frac{1}{4}}+\frac{1}{\frac{BY}{YE}+\frac{1}{4}}+\frac{1}{\frac{CZ}{ZF}+\frac{1}{4}}=

=\frac{1}{\frac{x(y+z)}{4yz}+\frac{1}{4}}+\frac{1}{\frac{y(z+x)}{4zx}+\frac{1}{4}}+\frac{1}{\frac{z(x+y)}{4xy}+\frac{1}{4}}=

=\frac{4yz}{yz+zx+xy}+\frac{4zx}{yz+zx+xy}+\frac{4xy}{yz+zx+xy}=4.