11.20*. From a point $M$ inside a given triangle $ABC$, perpendiculars $M A_{1}, M B_{1}, M C_{1}$ are dropped to the lines $BC, CA, AB$. For which points $M$ inside the given triangle $ABC$ does the value $a / M A_{1} + b / M B_{1} + c / M C_{1}$ attain its minimum?
11.20. Let \( x = M A_{1}, y = M B_{1} \) and \( z = M C_{1} \). Then \( a x + b y + c z = 2 S_{B M C} + 2 S_{A M C} + 2 S_{A M B} = 2 S_{A B C} \). Therefore, \(\frac{a}{x} + \frac{b}{y} + \frac{c}{z} \cdot 2 S_{A B C} = \frac{a}{x} + \frac{b}{y} + \frac{c}{z} \quad (a x + b y + c z) = a^{2} + b^{2} + c^{2} + a b \quad \frac{x}{y} + \frac{y}{x} + b c \quad \frac{y}{z} + \frac{z}{y} + a c \quad \frac{z}{x} + \frac{x}{z} \geqslant a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 a c\), and equality is achieved if and only if \( x = y = z \), i.e., \( M \) is the incenter of triangle \( A B C \).