No, these words do not share a common root, but they do all have something to do with feet. In fact, as English speakers we seem to be obsessed with feet. The words are everywhere! Let's start the investigation with, well, investigation. Did you know the root vestig comes from the Latin vestigium meaning "footprint, trace"? Yes, to investigate is etymologically to follow footprints!
And we're just getting started. What's up with pedigree? You may recognize ped, the most familiar foot root of all. For example, a millipede etymologically has a thousand feet, when you go on an expedition you are "freeing your feet" for adventure, an impediment is "that which blocks the feet," and a pedestrian is "one who is on foot." And as with most roots, a few alternative spellings have evolved. The words pawn, peon, and pioneer all map back to the Latin pedonem meaning "foot soldier." The word impeach comes from the Latin impedicare meaning "to fetter, catch, entangle"... you guessed it: the feet!
But let's walk this discussion back to pedigree. This comes from the French pié de grue, "foot of crane." Here's the story: back in the Middle Ages the symbol for "descent" was a forked sign resembling the branching lines of a genealogical chart. This sign also resembled a bird's footprint, specifically that of a crane. Eventually the whole subject of ancestry took on the name of this symbol. Learn more.
And speaking of lineage, poor Oedipus certainly had a lousy father! Who puts their baby up on a mountain with a stake through his feet just because some oracle is making crazy predictions? Too bad Oedipus wasn't a little older and stronger. Instead of ending up with "swollen feet" (the root meaning of Oedipus), he could have been recalcitrant. A new foot root to the rescue! This word comes from the Latin recalcitrare, "kicking back." Even better, the calc root means heel. So between kicking back and digging in his heels, Oedi could have saved his family a lot of trouble down the road. But then, it's likely the old man would have inculcated ("stamped in") his son with fear sooner or later, and driven him away.
But to where? Well, Patagonia would have been a great option! When life's complexities (as in Oedipal) have you down, it's time to head to what early European settlers called the "large foot" of South America! Learn more.