The path and the desire to reduce the suffering of others? It seems to me that the Noble Eight-fold Path is designed to reduce the suffering of the self.

But how do we use its guidance to reduce the suffering of others?

If the Buddha was walking down a road after enlightenment and saw
a strong man fighting a weaker man, or two equally strong men fighting, would he:


Desire to help one escape the other (and follow that desire)?
Desire to help one defeat the other (and follow that desire)?
Or (having no desire at all), keep on walking down the road?


And how should one who is still following The Path act in this situation?
By walking the path yourself you will be of great benefit to others. You are giving the gift of harmlessness.
A living example of the Dhamma, which is the path of complete peace and freedom.

The Buddha was a liberated one, a pure being, a great one. He no longer had any attachments whatsoever.
Helping others when asked or when a situation arises (proper, beneficial) is what pure beings do.

Having eradicated all mental defilements. Freed from avidity, anger, no longer having any kind of delusion, ignorance about the true nature of things.

Pure beings will never harm others, in any circumstances. Any possibilities to do so can no longer manifest themselves since they can no longer arise. Therefore they would never resort to violence, aggressive behavior and such.

What would the Blessed one do? I can only speculate for I am not such a great one...
It is said that the Buddha had supramundane powers. He knew exactly what was proper, as in the right course of action, what and when it was beneficial.

If reasonable and wise words could be beneficial in that particular instance, he would speak such words.
If not, he'd probably radiate metta ''goodwill'' and be on his way.