Are there any contemporary English words derived from same root as Euterpe & Terpsichore?
Euterpe & Terpsichore both contain the same etymological word root:
Euterpe muse of music, from Greek Euterpe, literally "well-pleasing," from eu "well" (see eu-) + terpein "to delight, please" (see Terpsichore). "
and
Terpsichore the muse of the dance, Greek Terpsikhore, literally "enjoyment of dance," from terpein "to delight" (from PIE root terp- "to satisfy;") + khoros "dance, chorus"
Does the Greek word root, terpein, or the parent PIE root terp- survive in any contemporary English word (not directly related to the two muses themselves)?
According to Wiktionary, thrive is from this root. (The OED does not go back beyond North Germanic for this, and Pokorny doesn't mention it).
thrive: from Middle English thriven, from Old Norse þrífa (“to seize, grasp, take hold, prosper”), from Proto-Germanic *þrībaną (“to seize, prosper”), from Proto-Indo-European *trep-, *terp- (“to satisfy, enjoy”).
Unfortunately, Wiktionary's policy on referencing only covers examples, not etymology, so there is no reference for the origin. The OED doesn't mention the "prosper" meaning for the Old Norse word,
The torpēre entry from Charlton T. Lewis, Charles Short, A Latin Dictionary begins:
torpĕo, ēre, v. n. Sanscr. root tarp-, to sate; Gr. τέρπω
I. to be stiff, numb, motionless, inactive, torpid, sluggish, etc. [...]
This is also mentioned at the Wiktionary entry for torpeo, French and English version. This would give torpor, for example. The English version goes on to suggest a link to PIE *ster (“stiff”), which projects into Old English as steorfan ("to die"), modern English starve.
Found en route starting from the Czech trpět ("to suffer"), by the way.
Edit. *(s)terp- entry in The tower of Babel by Sergei Starostin et al. mentions *terp- as possibly the same root.