Re: equilateral tringles

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
> C_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.

One can look upon each side (AB,BC,CA) as a vector cube root of unity,
e^i n 2 pi/3 for n = 0,1,2. The vector addition is such that (AB_1,
BC_1,CA_1) are formed by rotating each side by angle th giving m e^
i( n 2pi/3 + th) ( m < 1) which can be verified by cross product to
enclose an equilateral triangle for all th. A vector argument
concerning angles which gives a constant m is still needed here.

.

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