解：?$(1)$?∵?$a+b=2\sqrt {ab}$?
∴?$(\sqrt a-\sqrt b)^2=0$?
∴?$\sqrt a=\sqrt b，$?即?$a=b$?
∴原式?$=\frac {\sqrt {4a-a}}{\sqrt {5a+7a}}=\sqrt {\frac 3{12}}=\frac 12 $?
?$(2)$?∵?$a+b+c=2\sqrt {a-2}+4\sqrt {b-1}+6\sqrt {c+3}-14$?
∴?$(a-2)-2\sqrt {a-2}+1+b-1-4\sqrt {b-1}+4+c+3-6\sqrt {c+3}+9=0$?
∴?$(\sqrt {a-2}-1)^2+(\sqrt {b-1}-2)^2+(\sqrt {c+3}-3)^2=0$?
∴?$\sqrt {a-2}-1=0，$??$\sqrt {b-1}-2=0，$??$\sqrt {c+3}-3=0$?
∴?$a=3，$??$b=5，$??$c=6$?
