Re: equilateral tringles
- From: "Philippe 92" <nospam@xxxxxxxxxxxx>
- Date: Thu, 01 Dec 2005 00:57:15 +0100
eugene 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.
>
> I'd be very grateful if someone could help me.
In complex plane :
A triangle abc is equilateral (direct) iff a + j*b + j^2*c = 0
with j = exp(i*pi/3) = (1 + i*sqrt(3))/2
That is from vector AC = AB*exp(i*pi/3), and AC = c-a etc...,
using j^2 + j + 1 = 0, because j^3 - 1 = (j-1)(j^2 + j + 1) = 0.
In this problem we have :
a1 = (c1 + c)/2 etc...
Calculate a1 + j*b1 + j^2*c1, use abc equilateral and j^3 = 1,
conclude.
Regards.
--
philippe
mail : chephip at free dot fr
site : http://chephip.free.fr/
.
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