If α, β, γ are the angles of a triangle and the system of equationscos⁡(α−β)x+cos⁡(β−γ)y+cos⁡(γ−α)z=0cos⁡(α+β)x+cos⁡(β+γ)y+cos⁡(γ+α)z=0sin⁡(α+β)x+sin⁡(β+γ)y+sin⁡(γ+α)z=0has non-trivial solutions, then triangle is necessarily

Let Δ=cos⁡(α−β)cos⁡(β−γ)cos⁡(γ−α)cos⁡(α+β)cos⁡(β+γ)cos⁡(γ+α)sin⁡(α+β)sin⁡(β+γ)sin⁡(γ+α)It is clear that either α=β or β=γ or γ=α s sufficient to make ∆=0. It is not necessary that triangle is equilateral.Also, isosceles triangle can be obtuse one.