Question

求解几何题

Answer 1

面积法算不算纯几何?
连FD, DE, EG.
∵S△ADF/S△ABC = AD/AB·AF/AC = 1/6,
S△DBE/S△ABC = BE/BC·BD/BA = 1/8,
S△FEC/S△ABC = CF/CA·CE/CB = 1/2,
∴S△DEF = S△ABC-S△ADF-S△DBE-S△FEC = (1-1/6-1/8-1/2)S△ABC = 5/24·S△ABC.
又∵S△GEC/S△ABC = CG:CA·CE/CB = 1/4,
∴S△GEF = S△FEC-S△GEC = (1/2-1/4)S△ABC = 1/4·S△ABC,
∴DH:HG = S△DEF:S△GEF = 5:6 ①.
类似的, ∵S△ADG/S△ABC = AD/AB·AG/AC = 1/3,
∴S△EDG = S△ABC-S△ADG-S△DBE-S△GEC = (1-1/3-1/8-1/4)S△ABC = 7/24·S△ABC,
S△FDG = S△ADG-S△ADF = (1/3-1/6)S△ABC = 1/6·S△ABC,
∴EH:HF = S△EDG:S△FDG = 7:4 ②.

注: 由上面过程可知, △ABC等边的条件是多余的.
另外, 比面积法更"纯"的几何方法也有(但是面积法辅助线相对简单),
这里只说大意: 过D作DN // BC交EF于M, 交AC于N, 再过G作GP // BC交EF于P.
求出DN/BC = 1/2, MN/BC = 3/16而得到DM/BC = 5/16, 又可求出GP/BC = 3/8.
最后可求得DH:HG = DM:GP = 5:6, 对EH:HF类似.