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In geometry, the Fermat point of a triangle, also called the Torricelli point or Fermat-Torricelli point, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible.〔(Cut The Knot - The Fermat Point and Generalizations )〕 It is so named because this problem is first raised by Fermat in a private letter to Evangelista Torricelli, who solved it. The Fermat point gives a solution to the geometric median and Steiner tree problems for three points. == Construction == The Fermat point of a triangle with largest angle at most 120° is simply its first isogonic center or X(13), which is constructed as follows: # Construct an equilateral triangle on each of two arbitrarily chosen sides of the given triangle. # Draw a line from each new vertex to the opposite vertex of the original triangle. # The two lines intersect at the Fermat point. An alternate method is the following: # On each of two arbitrarily chosen sides, construct an isosceles triangle, with base the side in question, 30-degree angles at the base, and the third vertex of each isosceles triangle lying outside the original triangle. # For each isosceles triangle draw a circle, in each case with center on the new vertex of the isosceles triangle and with radius equal to each of the two new sides of that isosceles triangle. # The intersection inside the original triangle between the two circles is the Fermat point. When a triangle has an angle greater than 120°, the Fermat point is sited at the obtuse-angled vertex. In what follows "Case 1" means the triangle has an angle exceeding 120°. "Case 2" means no angle of the triangle exceeds 120°. 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「Fermat point」の詳細全文を読む スポンサード リンク
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