Pigeonhole principle for a triangle 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 $\frac 15$.
I have seen answers with 9 points (in comment link above) and 5 points (above). 

However, not with 7 points. Please see below:

Drawlines from midpoints of all 3 sides to each other

Now we have 6 points (3 vertices and 3 mid points)

we have 4 triangles of equal area 1/4. (equilateral triangle)

To add the 7th point will create a triangle of area less than 1/4.

QED