$[\underset{\text { The ratio of areas of triangles with a common angle }}{[\quad \text { Classical inequalities (other) }}]$

On the hypotenuse $AB$ of a right triangle $ABC$, a point $X$ is taken, and $M$ and $N$ are its projections on the legs $AC$ and $BC$.

a) For what position of point $X$ will the length of segment $MN$ be the smallest?

b) For what position of point $X$ will the area of quadrilateral $CMXN$ be the largest?
a) Since $C M X N$ is a rectangle, then $M N = C X$. Therefore, the length of the segment $M N$ will be the smallest if $C X$ is the height.

b) Let $S_{A B C} = S$. Then $S_{A M X} = \frac{A X^{2} \cdot S}{A B^{2}}$ and $S_{B N X} = \frac{B X^{2} \cdot S}{A B^{2}}$. Since $A X^{2} + B X^{2} \geq 1 / 2 A B^{2}$ (with equality only when $X$ is the midpoint of segment $A B$), then $S_{C M X N} = S - S_{A M X} - S_{B N X} \leq S / 2$.

## Answer

a) When $X$ is the foot of the height.

b) When $X$ is the midpoint of segment $A B$.