Geometric Inequalities
Geometric Inequalities
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10.1142/9789812709431_0007
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符号和基本事实a、b、c分别是∆ABC与a、b、c相对的边。(ABC) =∆ABC面积s = semi-perimeter =) c b (2 1 + + r =内接圆半径r =外接圆半径正弦规则:r b 2 c罪c b罪一罪= = = cos规则:一个公元前2 = b + c 2−2 cos (ABC) = b罪ac 2 1赎罪公元前2 1 c罪ab ABC =) 2 1 = = = r 4 c s) (b s) (a s (s−−−(海伦的公式)= 2 cr br 2 ar + + = sr示例1等周定理在所有三角形与三角形一个固定的周长,等边三角形最大的区域。证明:使用鹭的公式和AM-GM不等式(ABC) = 3 3 3 s年代3)c s b () () (s) c年代)(b s) (a s (s 2 3 3 ==−+−−≤−−−与平等拥有当且仅当年代−=−b =年代−c,即a = b = c。示例2(1961年国际海事组织)让一个b, c是一个三角形的边,和T它的面积。证明：a 2 + b 2 + c 2≥4 3 t，在什么情况下等式成立？解决方案1:p表示三角形的周长,即p = a + b + c,等周定理的三角形,我们有T≤4 3 3 p 2与平等拥有当且仅当a = b = c - (1) cauchy - schwarz不等式给p 2 = (a + b + c) 2≤3 c b(2 + 2 + 2)与平等拥有当且仅当a = b = c -(2)它遵循从(1)和(2)T≤4 2 2 3 3 c b 2 b + +相当于2 + 2 + c 2≥4 3 T与平等拥有当且仅当a = b = c。解决方案2:边长为c的等边三角形的高度为23…
Notation and Basic Facts a, b, and c are the sides of ∆ABC opposite to A, B, and C respectively. [ABC] = area of ∆ABC s = semi-perimeter =) c b a (2 1 + + r = inradius R = circumradius Sine Rule: R 2 C sin c B sin b A sin a = = = Cosine Rule: a 2 = b 2 + c 2 − 2bc cos A [ABC] = B sin ac 2 1 A sin bc 2 1 C sin ab 2 1 = = = R 4 abc =) c s)(b s)(a s (s − − − (Heron's Formula) = 2 cr 2 br 2 ar + + = sr Example 1 Isoperimetric Theorem for Triangle Among all triangles with a fixed perimeter, the equilateral triangle has the largest area. Proof: Using the Heron's Formula and the AM-GM inequality [ABC] = 3 3 s 3 s s 3) c s () b s () a s (s) c s)(b s)(a s (s 2 3 3 = = − + − + − ≤ − − − with equality holds if and only if s − a = s − b = s− c, i.e a = b = c. Example 2 [IMO 1961] Let a, b, c be the sides of a triangle, and T its area. Prove: a 2 + b 2 + c 2 ≥ 4 3 T. In what case does equality hold? 1st solution: Denote the perimeter of the triangle by p, i.e p = a + b + c, by the isoperimetric theorem for triangle, we have T ≤ 4 3 3 p 2 with equality holds if and only if a = b = c-(1) The Cauchy-Schwarz inequality gives p 2 = (a + b + c) 2 ≤ 3(a 2 + b 2 + c 2) with equality holds if and only if a = b = c.-(2) It follows from (1) and (2) that T ≤ 4 3 3 c b a 2 2 2 + + which is equivalent to a 2 + b 2 + c 2 ≥ 4 3 T with equality holds if and only if a = b = c. 2nd solution: An equilateral triangle with side c has altitude 2 3 …