Re: [幾何] 正三角形內部任兩點距離小於邊長

[JohnMash (Paul)]

※ 引述《cnick (cnick)》之銘言：
: 正三角形內部任兩點距離小於邊長


     ΔABC is a right triangle with side length a

     and D,E are interior points

     (i) |AD|,|AE| < a

     (ii) ∠DAE < π/3

     then ∠ADE or ∠DEA > π/3

     then |DE| < max{|AD|,|AE|} < a

------------------------------

     proof of (i)

     plot line AD intersecting BC at F

     then |AF| > |AD|

     and ∠AFC + ∠CAF = 2π/3

     and ∠CAF < π/3

     hence, ∠AFC > π/3

     hence, a = |AC| > |AF| > |AD|

     similarly, a > |AE|



--
※ 發信站: 批踢踢實業坊(ptt.cc) 
◆ From: 123.194.224.241
推 cnick       :謝謝