Midpoint Link Theorem

A theorem that states that a line segment connecting the midpoints of two sides of a triangle is parallel to the third side and is equal to half its length. The converse of this theorem is also true: "A line segment that passes through the midpoint of one side of a triangle and is parallel to the other side passes through the midpoint of the third side." Applications of this theorem include "the midpoint of the hypotenuse of a right triangle is equidistant from the three vertices," "connecting the midpoints of the three sides of a triangle divides the triangle into four congruent triangles," "connecting the midpoints of the four sides of a quadrilateral creates a parallelogram," and "two line segments connecting the midpoints of opposite sides of a quadrilateral bisect each other."

[Toshio Shibata]

Source: Shogakukan Encyclopedia Nipponica About Encyclopedia Nipponica Information | Legend

三角形の二辺の中点を結ぶ線分は第三辺に平行で長さはその半分に等しい、という定理。この定理の逆の一つで、「三角形の一辺の中点を通り他の一辺と平行な直線は第三辺の中点を通る」も成立する。この定理の応用として、「直角三角形の斜辺の中点は三頂点から等距離にある」「三角形の三辺の中点を結ぶことにより三角形は四つの合同な三角形に分けられる」「四角形の四辺の中点を結ぶと平行四辺形ができる」「四辺形の対辺の中点を結ぶ二つの線分は互いに他を二等分する」などがある。

［柴田敏男］

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