Consider a equilateral triangle of total area 1. Suppose 7 points are chosen inside. Show that some 3 points form a triangle of area $\leq\frac 14$.
Choose one point $p$ and draw a line from $p$ to each of the other six points.
We can order the six points $a,b,c,d,e,f$ going clockwise around point $p$ and draw lines from $a$ to $b$, from $b$ to $c$, from $c$ to $d$, from $d$ to $e$ and from $e$ to $f$.
This gives five disjoint triangles which fit inside the triangle of area $1$.
Therefore the total area of the five triangles is less than or equal to one and at least one triangle has area less than or equal to $\frac 15$.