AN IDENTITY

Consider three lines,
*L*_{1}
*L*_{2}
*L*_{3},
passing through a point *P*_{0},
and pick three arbitrary points, one on each line,
different from *P*_{0}:
*P*_{1} on *L*_{1},
*P*_{2} on *L*_{2},
*P*_{3} on *L*_{3}.

Consider the lines joining these three points:
*L*_{12} joining *P*_{1} and *P*_{2},
*L*_{23} joining *P*_{2} and *P*_{3},
*L*_{31} joining *P*_{3} and *P*_{1}.

Let *a*_{1} be the angle between
*L*_{1} and *L*_{12},
and *b*_{1} the angle between
*L*_{1} and *L*_{31}.
Similarly let *a*_{2} be the angle between
*L*_{2} and *L*_{12},
and *b*_{2} that with
*L*_{23}.
Finally let *a*_{3} be the angle between
*L*_{3} and *L*_{31},
and *b*_{3} that with
*L*_{23}.

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(*a*_{1}) sin(*a*_{2})
sin(*a*_{3}) = sin(*b*_{1})
sin(*b*_{2}) sin(*b*_{3})

**Hint**

Use the law of sines.

Marco Corvi - 2005