Exercise 1095Combinatorics › Pigeonhole principleOlympiadAdded 3 Nov 2026

Points in an equilateral triangle

#pigeonhole#equilateral#distance

Statement

Show that among any $5$ points in an equilateral triangle of side $1$ , two are at distance at most $\frac12$ .

Hint

Divide the triangle into $4$ equilateral triangles of side $1/2$ .

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Subdivision. Connect the midpoints of the sides to get $4$ equilateral triangles of side $1/2$ .

Pigeonhole. $5$ points in $4$ subtriangles: some contains at least $2$ points.

Distance. Diameter of a side- $(1/2)$ equilateral triangle is $1/2$ .

Conclusion. Two points are at distance $\le 1/2$ .