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:''For the basic inequality'' ''a'' < ''b'' + ''c'', ''see Triangle inequality.'' :''For inequalities of acute or obtuse triangles, see Acute and obtuse triangles.'' In geometry, triangle inequalities are inequalities involving the parameters of triangles, that hold for every triangle. The inequalities give an ordering of two different values: they are of the form "less than", "less than or equal to", "greater than", or "greater than or equal to". The parameters in a triangle inequality can be the side lengths, the semiperimeter, the angle measures, the values of trigonometric functions of those angles, the area of the triangle, the medians of the sides, the altitudes, the lengths of the internal angle bisectors from each angle to the opposite side, the perpendicular bisectors of the sides, the distance from an arbitrary point to another point, the inradius, the exradii, the circumradius, and/or other quantities. Unless otherwise specified, this article deals with triangles in the Euclidean plane. ==Main parameters and notation== The parameters most commonly appearing in triangle inequalities are: *the side lengths ''a'', ''b'', and ''c''; *the semiperimeter ''s'' = (''a'' + ''b'' + ''c'') / 2 (half the perimeter ''p''); *the angle measures ''A'', ''B'', and ''C'' of the angles of the vertices opposite the respective sides ''a'', ''b'', and ''c'' (with the vertices denoted with the same symbols as their angle measures); *the values of trigonometric functions of the angles; *the area ''T'' of the triangle; *the medians ''m''''a'', ''m''''b'', and ''m''''c'' of the sides (each being the length of the line segment from the midpoint of the side to the opposite vertex); *the altitudes ''h''''a'', ''h''''b'', and ''h''''c'' (each being the length of a segment perpendicular to one side and reaching from that side (or possibly the extension of that side) to the opposite vertex); *the lengths of the internal angle bisectors ''t''''a'', ''t''''b'', and ''t''''c'' (each being a segment from a vertex to the opposite side and bisecting the vertex's angle); *the perpendicular bisectors ''p''''a'', ''p''''b'', and ''p''''c'' of the sides (each being the length of a segment perpendicular to one side at its midpoint and reaching to one of the other sides); *the lengths of line segments with an endpoint at an arbitrary point ''P'' in the plane (for example, the length of the segment from ''P'' to vertex ''A'' is denoted ''PA'' or ''AP''); *the inradius ''r'' (radius of the circle inscribed in the triangle, tangent to all three sides), the exradii ''r''''a'', ''r''''b'', and ''r''''c'' (each being the radius of an excircle tangent to side ''a'', ''b'', or ''c'' respectively and tangent to the extensions of the other two sides), and the circumradius ''R'' (radius of the circle circumscribed around the triangle and passing through all three vertices). 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「List of triangle inequalities」の詳細全文を読む スポンサード リンク
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