In an equilateral triangle, mark every point whose distance from the nearest vertex is less than the distance from the centroid. What percentage of the area of the equilateral triangle is occupied by points with the mentioned property?

(L. Šimůnek)
First, we construct the perpendicular bisectors of the segments, one of whose endpoints is the centroid and the other is the vertex of the given triangle. Since the perpendicular bisector of a segment is the set of points that are equidistant from both endpoints of the segment, the points in the triangle that satisfy this condition always lie in the half-plane containing the corresponding vertex of the triangle (fig.).
![](https://cdn.mathpix.com/cropped/2024_04_17_e907aea6a48a86a4c76eg-3.jpg?height=474&width=1198&top_left_y=234&top_left_x=457)

An equilateral triangle can be divided into 9 congruent equilateral triangles (fig.), with three of them corresponding to the points with the given property. This means that these points occupy $33 \frac{1}{3} \%$ of the area of the triangle.