Dear Greg and Alex,
We can study some other interesting facts related "A Relation in Triangle". For convenience I use some detail notations:
A triangle ABC cuts off from the circumcircle three circular segments. Let A', B', C' are midpoints of these circular segments with respect to BC, CA, AB respectively.
k, l, m are distances from A', B', C' to sidelines BC, CA, AB respectively.
Greg's result: 2*k*l*m = R*r^2
where R is the circumradius, r is the inradius.
Now let p, q, r are distances from A, B, C to sidelines B'C', C'A', A'B' respectively then we can have similar result: 2*p*q*r = R*r^2
This result can be derivered from one more general following fact (may be new):
Let A, B, C, A', B', C' are any six concyclic points
k, l, m are distances from A', B', C' to lines BC, CA, AB respectively.
p, q, r are distances from A, B, C to lines B'C', C'A', A'B' respectively.
The result: k*l*m = p*q*r
I hope that we can find nice synthetic proof for this fact.
Best regards,
Bui Quang Tuan