Concurrent lines

A triangle ABC has the following dimensions :
- BC = a = 1033
- AC = b = 1553
- AB = c = 1973

We choose 3 points:
- P on AB at a distance p of A (0 < p < c)
- Q on BC at a distance q of B (0 < q < a)
- R on AC at a distance r of C (0 < r < b)
(p,q,r are integers)

How many triples (p,q,r) are there such as the lines AQ, BR and CP are concurrent?

Answer format: $\text{Count},\sum p,\sum q,\sum r$

You are given : 4,214,202,206 for the triangle 101,103,107

[My timing: < 20 sec]