Crea sito
AN IDENTITY


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

sin(a1) sin(a2) sin(a3) = sin(b1) sin(b2) sin(b3)

Hint
Use the law of sines.

Marco Corvi - 2005