以線段AD為直徑作圓,B、C把線短AD三等分,在圓上取一點E,(但E≠A,E≠D),令角AEB=α,角CED=β,tanα×tanβ?
考慮ABE
AB/sinα=EB/sinEAB
EDC
CD/sinβ=EC/cosEAB
相乘得
AB CD/sinαsinβ = EB EC /sinEAB cosEAB...(1)
考慮ACE
AC/cos β = EC/sinEAB
BDE
BD/cos α = EB/cosEAB
相乘得
AC BD/cosαcosβ=EB EC/sinEAB cosEAB ...(2)
注意(1),(2)右邊相等,故
AB CD/sinαsinβ =AC BD/cosαcosβ
tanα anβ=AB CD/ AC BD = 1/4