Triangle area via pigeonhole

Subsquares. Divide into $4$ subsquares of side $1/2$: some subsquare contains $2$ points $A,B$.

Triangle with another point. Take any third point $C$ among the remaining $3$. The triangle $ABC$ has base $AB\le\sqrt2/2$ and height $\le\sqrt2$ (the diagonal of the square). So area $\le\frac12\cdot\frac{\sqrt2}{2}\cdot\sqrt2=\frac12$.

Conclusion. Three points form a triangle of area $\le 1/2$.