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?
Do not write an equation yet. Shake the figure first.
Play the reversal slowly
Now move the same pieces step by step and prove that the pattern you touched was not an accident.
A triangle. Draw the three medians (vertex to opposite-side midpoint).
Return to the exact question
Compare the areas of the six small triangles formed by a triangle’s three medians.
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.
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