The solution to Putnam 2000 A5 uses this formula, for which the following proof is given: (source: https://mks.mff.cuni.cz/kalva/putnam/psoln/psol005.html)
Let the sides (of triangle ABC) have lengths a, b, c as usual. The question suggests that we use some relationship of the form abc = constant x R. ...
To prove the relation, let O be the centre of the circumcircle. Project AO to meet the circle again at K. Let AH be the altitude. Then angle ABC = angle AKC, so triangles ABH and AKC are similar. Hence AB/AH = AK/AC or c/AH = 2R/b. Hence abc = 2R·a·AH = 4ΔR.
The bold part confuses me. How does ABC = AKC? K is dependent wholly on O and A whereas B is independent of both. And if ABC = AKC, how does that lead to ABH ~ AKC?