Question

求证一道几何题

如何用几何方法证明？

Answer 1

BD/AD = tan∠BAD
BC/AC = 1/tan∠ABC
记 tan∠BAD = t, 则根据3×BD/AD = BC/AC 得到tan∠ABC = 1/3t

CE/OC = sin∠AOC = sin(2∠ABC) = (2/3t)/(1+1/(3t²)) ......三角函数万能公式
BF/OF = sin∠BOD = sin(2∠BAD) = 2t/(1+t²) ......三角函数万能公式
同样再利用三角函数公式
cos∠AOC = cos(2∠ABC) = [1-(1/3t)²]/[1+(1/3t)²]
cos∠BOD = cos(2∠BAD) = (1-t²)/(1+t²)

2×CG/OC = 2sin∠COD = 2sin(∠AOC+∠BOD) = 2(sin∠AOC×cos∠BOD + cos∠AOCsin∠BOD)
=...化简一下 = (24t³+8t)/[(1+t²)(1+9t²)] ........等式的右边

CE/OC+BF/OF = ...化简一下 = (24t³+8t)/[(1+t²)(1+9t²)] .......等式的左边
两者恰好相等，因而得证。