Re: equilateral tringles

Dave Rusin wrote :
> In article <mn.08487d5c306dddd6.22155@xxxxxxx>,
> Philippe 92 <nospam@xxxxxxxxxxxx> wrote:
>> Dave Rusin wrote :
>>> In article <3950172.1133377985893.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
>>> eugene <jane1806@xxxxxxx> wrote:
>>>
>>>> Let ABC be an equilateral triangle. Points A_1,B_1,C_1 are chosen
>>> inside the triangle in such a way that A_1 \in CC_1, B_1 \in AA_1,
>>> C_1 \in BB_1 and AB_1=B_1A_1, BC_1=C_1B_1, CA_1=C_1A_1. Prove that
>>> the triangle A_1B_1C_1 is also equilateral.
>
>>> The word "chosen" is misleading here: the points A1, B1, C1 are
>>> uniquely specified by the condition that each of them is a midpoint
>>> of one of these three segments (and an endpoint of another).
>>> But then, that means that a one-third turn of the figure preserves
>>> all the interior line segments,
>>
>> Why that ? Yes if AA_1 = BB_1 = CC_1, or if angles are equal...
>> that is if A_1B_1C_1 equilateral, you proove that it is equilateral !
>
> No -- actually I think your question is inserted in the wrong space.
> So really your question "Why that?"
> should be two lines earlier, when I asserted uniqueness.

I agree : prooving the uniqueness results into symetry.
Uniqueness is easy to proove however as you said.
My approach discarded completely solving these equations, but just
proove directly that the inner triangle is equilateral, whatever it
could be. Hence my "blindness".

Regards.

--
philippe
mail : chephip at free dot fr
site : http://chephip.free.fr/


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