Consider three lines, L₁ L₂ L₃, passing through a point P₀, and pick three arbitrary points, one on each line, different from P₀: P₁ on L₁, P₂ on L₂, P₃ on L₃.

Consider the lines joining these three points: L₁₂ joining P₁ and P₂, L₂₃ joining P₂ and P₃, L₃₁ joining P₃ and P₁.

Let a₁ be the angle between L₁ and L₁₂, and b₁ the angle between L₁ and L₃₁. Similarly let a₂ be the angle between L₂ and L₁₂, and b₂ that with L₂₃. Finally let a₃ be the angle between L₃ and L₃₁, and b₃ that with L₂₃.

This is a scketch of the geometry:

               |           |   
            \b1|           |b2/
             \ | a1        | / 
              \|           |/ a2
             P1+-----------+----- l12
               |\         /|P2
               | \       / |
               |  \l31  /  |
               |   \   /   |
               |    \ /    |
               |     X     |l23
               |    / \    |
             l1|   /   \   |
               |  /     \  |
               | /l2     \ |
               |/         \| b3
               +-----------+-----
             P0     l3    P3\ a3
                             \

Then the following identity holds

Hint
Use the law of sines.

Marco Corvi - 2005