The events are not independent. Therefore:
P(A wins) = P(A beats B) * P(A beats C | A beats B)
So, the question becomes, "How do you calculate the conditional probability?"
I think that if you divide each result by their sum, you get the right values.
P(A wins) = .286/.788
P(B wins) = .096/.788
P(C wins) = .406/.788
To convince yourself that this makes some sense, set all probabilities to .5. Then each runner's probability of winning (prior to dividing by the sum) is 0.25. After dividing by the sum, the probabilities become 1/3, which is what you'd expect.