■直観幾何学研究会(その6)

 (その5)の続き.

  cos5φ=a/2R

  cos3φ=b/2R

  cosφ=c/2R

  cosπ/7=2cos^2π/14−1=c^2/2R^2−1

  cos3π/7=2cos^23π/14−1=b^2/2R^2−1

  cos5π/7=2cos^25π/14−1=a^2/2R^2−1

  cosπ/7+cos3π/7+cos5π/7=1/2

=(a^2+b^2+c^2)/2R^2−3=1/2

  a^2+b^2+c^2=−5R^2

  cosπ/7・cos3π/7+cos3π/7・cos5π/7+cos5π/7・cosπ/7=−1/2

  (c^2/2R^2−1)(b^2/2R^2−1)+(b^2/2R^2−1)(a^2/2R^2−1)+(a^2/2R^2−1)(c^2/2R^2−1)=−1/2

  (c^2−2R^2)(b^2−2R^2)+(b^2−2R^2)(a^2−2R^2)+(a^2−2R^2)(c^2−2R^2)=−2R^4

  a^2b^2+b^2c^2+c^2a^2−4R^2(a^2+b^2+c^2)+12R^4=−2R^4

  a^2b^2+b^2c^2+c^2a^2=−34R^4

  cosπ/7・cos3π/7・cos5π/7=−1/8

  (c^2/2R^2−1)(b^2/2R^2−1)(a^2/2R^2−1)=−1/8

  (c^2−2R^2)(b^2−2R^2)(a^2−2R^2)=−R^6

  a^2b^2c^2−2R^2(b^2c^2+c^2a^2+a^2b^2)+4R^4(a^2+b^2+c^2)−8R^6=−R^6

  a^2b^2c^2+68R^6−20R^6−8R^6=−R^6

  a^2b^2c^2=−41R^6

===================================

  1/a^2+1/b^2+1/c^2

=(a^2b^2+b^2c^2+c^2a^2)/a^2b^2c^2=34/41R^2≠2/R^2

 計算ミスがあるようだが,本日は時間切れ.

===================================