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The right triangle altitude theorem or geometric mean theorem is a result in elementary geometry that describes a relation between the altitude on the hypotenuse in a right triangle and the two line segments it creates on the hypotenuse. It states that the geometric mean of the two segments equals the altitude. ==Theorem and application== If ''h'' denotes the altitude in a right triangle and ''p'' and ''q'' the segments on the hypotenuse then the theorem can be stated as: : or in term of areas: : The latter version yields a method to square a rectangle with ruler and compass, that is to construct a square of equal area to a given rectangle. For such a rectangle with sides ''p'' and ''q'' we denote its top left vertex with ''D''. Now we extend the segment ''q'' to its left by ''p'' (using arc ''AE'' centered on ''D'') and draw a half circle with endpoints ''A'' and ''B'' with the new segment ''p+q'' as its diameter. Then we erect a perpendicular line to the diameter in ''D'' that intersects the half circle in ''C''. Due to Thales' theorem ''C'' and the diameter form a right triangle with the line segment ''DC'' as its altitude, hence ''DC'' is the side of a square with the area of the rectangle. The converse statement is true as well. Any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle. Historically the theorem is attributed to Euclid (ca. 360–280 BC), who stated it as a corollary to proposition 8 in book VI of his Elements and used it in proposition 14 of book II to square a rectangle. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Geometric mean theorem」の詳細全文を読む スポンサード リンク
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