Answer: 364

Small Hint:

Apply the third property to $(x,y-x)$ when $y\gt x$

Big Hint:

Compare the resulting subtraction rule with the same rule for $\operatorname{lcm}(x,y)$

Solution:

For $y\gt x,$ the third property applied to $(x,y-x)$ gives $$f(x,y)=\frac{y}{y-x}f(x,y-x).$$ The least common multiple obeys the identical relation, because $\gcd(x,y)=\gcd(x,y-x).$ Hence the quotient $\frac{f(x,y)}{\operatorname{lcm}(x,y)}$ is unchanged by each subtraction step of the Euclidean algorithm. On the diagonal it equals $\frac{f(g,g)}{g}=1,$ so $f(x,y)=\operatorname{lcm}(x,y).$ Therefore $$\begin{aligned}f(14,52)&=\frac{14\cdot52}{\gcd(14,52)}\\&=364.\end{aligned}$$