$證明：∵?△ABC\sim △A'B'C'?$
$∴?∠B=∠B'，??\frac {AB}{A'B'}=\frac {BC}{B'C'}=k?$
$∵?AD?與?A'D'?分別是?△ABC?和?△A'B'C'?中邊?BC、??B'C'?上的中線$
$∴?BC= 2BD，??B'C'=2B'D'?$
$∵?\frac {BC}{B'C'}=k?$
$∴?\frac {2BD}{2B'D'}=k?$
$∴?\frac {BD}{B'D'}=k?$
$∴?\frac {AB}{A'B'}=\frac {BD}{B'D'}=k?$
$∴?△ABD∽△A'B'D'?$
$∴?\frac {AB}{A'B'}=\frac {AD}{A'D'}=k?$
