We have a triangle ABC, with a point D on side AB。。。【海口学习吧】

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We have a triangle ABC, with a point D on side AB。。。

We have a triangle ABC, with a point D on side AB, E, on side BC, and F on side CA. The smaller triangle, DEF, is equilateral. The line segments AD, BE, and CF all have equal length.
Problem: Prove that ABC is also equilateral.

送TA礼物

1楼2012-08-21 14:59回复

Assume triangle ABC is not equilateral,
So evidently we can assume ∠B <60< ∠A.
now we can find M,N on side AB in order to make ∠FND=∠DME=60°
∵ ∠NDE=∠DEM+∠DME，
∠FDE=∠DEF=∠EFD=60°=∠DME
∴∠DEM=∠NDF.
and DF=DE
∴△FDN≌△DEM(AAS)
∴DN=EM
∵∠EMB=120°
∴BE>EM
but according to the known conditions ：AD=BE
it is impossible to make AD+AN=DN=EN(BE>EM).
so triangle ABC is also equilateral.