First slide

System of linear equations

Question

If $α, β, γ$ are the angles of a triangle and the system of equations
$\begin{array}{l} \cos (α - β) x + \cos (β - γ) y + \cos (γ - α) z = 0 \\ \cos (α + β) x + \cos (β + γ) y + \cos (γ + α) z = 0 \\ \sin (α + β) x + \sin (β + γ) y + \sin (γ + α) z = 0 \end{array}$
has non-trivial solutions, then triangle is necessarily

Moderate

A

equilateral
B

isosceles
C

right angled
D

acute angled

Solution

Let $Δ = |\begin{array}{ccc} \cos (α - β) & \cos (β - γ) & \cos (γ - α) \\ \cos (α + β) & \cos (β + γ) & \cos (γ + α) \\ \sin (α + β) & \sin (β + γ) & \sin (γ + α) \end{array}|$
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.

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