Geoffrey:
Sounds good but I haven't taken the time to run through a load of examples.
And I think if you do, you will find that the French teacher is absolutely correct. I can't think of any example where we'd pronounce it 'the', rather than 'thee', before a word beginning with a vowel.
The vowel sound in 'the' is, I believe, called the 'schwa' - it's our most common sound, also our laziest: open your mouth and make the easiest, grunt-like sound that comes and that's the schwa. In multi-syllabic words, we tend to stress one syllable in particular (e.g. reMARKable; aMAZing; CORPoral PUNishment) with the result that we are lazier with the vowel sounds in the non-stressed syllables - which are often the schwa (eg. the 'ab' in remarkable; the 'a' at the start of amazing, the 'or' in corporal and the 'ment' at the end of punishment). For some reason, it sounds and feels awkward to follow the schwa with another vowel sound, hence we came up with the longer vowel sound of 'thee'.
Talking of extra Es, but on a different note entirely, here's a true story from my original home county of Yorkshire, told to me many years ago by my always-truthful, dearly-departed father:
A recently bereaved widower ordered a gravestone for his late wife, who'd been a devout Christian. In recognition of her devotion, the epitaph on the stone should, he said, read simply: "SHE WAS THINE".
Summoned a few days later to approve the finished article, he was bewildered to see that it said: "SHE WAS THIN".
"You daft bugger!" he berated the stonemason. "Tha's left off the E!"
Promising to put it right, the man told him to return later that afternoon. But when he did, the widower was more bewildered than ever, for the gravestone now bore the legend:
"E, SHE WAS THIN"