ILL 1978 Problem 53

Determine all the triples $(a, b, c)$ of positive real numbers such that the system $$ax + by -cz = 0,$$ $$a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,$$

is compatible in the set of real numbers, and then find all its real solutions.