Two chords $$AB$$ and $$CD$$ of a circle are intersecting in a point $$P$$. A relation between the lengths of all the line segments $$PA$$, $$PB$$, $$PC$$ and $$PD$$ is __________

PLACEHOLDER ANSWER - the answer is a relation, not a number. The required relation is $$PA \cdot PB = PC \cdot PD$$. In triangles $$PAC$$ and $$PDB$$, $$\angle APC = \angle DPB$$ (vertically opposite) and $$\angle PAC = \angle PDB$$ (angles in the same segment on chord $$BC$$), so the triangles are similar. Hence $$\frac{PA}{PD} = \frac{PC}{PB}$$, which gives $$PA \cdot PB = PC \cdot PD$$.