Three Medians, Six Equal Areas
A triangle’s three medians split it into six small triangles. Different shapes — yet all six have equal area. Why?
Problem
Compare the areas of the six small triangles formed by a triangle’s three medians.
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A triangle. Draw the three medians (vertex to opposite-side midpoint).
Why this approach works
A median joins a vertex to the midpoint of the opposite side; the three meet at the centroid G. Key fact: a median splits a triangle into two equal areas (same height, half the base). Each median halves, and the next halves again, so the six small triangles all have equal area (⅙ of the whole). That balance is why G is the center of gravity.
Proof
A median halves the area (half the base, shared height). Three medians halving in turn make six triangles each ⅙ of the whole — which is why the centroid is the balance point.
Try it yourself
Draw just one median: it halves the triangle (half the base, shared height). Three such halvings overlap into six equal parts.
Related mathematics
Beyond MathVoyage
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