Newton: Pride cometh before a fall

I found a draft of this while looking through my old posts, and realized I had written it four months ago and forgotten to post it.
Here you go:

Yesterday was a triumphant day in lab class.
One of my classmates, who has a reputation for thinking that he knows better than the authors we read what they ought to write, and is fond of attempting to correct the argument of Aquinas, was doing a prop in lab. We've gotten used to his overbearing manner, but this time it was insufferable. It might have been that he was showing off to his little brother, who was sitting in on the class, but his attitude was driving the rest of us into the wall, by constantly using such phrases as, "this is so obvious I don't even know why he bothers to say it," "It is manifest that," and "It's the most obvious thing possible." Our tutor, Dr. Decaen, at one point asked him about the third line from the bottom. The student answered a question about the fourth line from the bottom, whereupon Dr. Decaen pointed out that he had asked about the third line, although the student was right, and the actual question had to do with the fourth line. Whereupon our villain responded, "Well, I thought you couldn't be asking about the line, because only an idiot would question that." At this point, the entire class was just shocked at his pompous, arrogant manner, too stunned to reply.
Next, he mentioned that, "while Newton's way of proving this works, I've devised a better and simpler way of doing it. Here, I'll show you." And he proceeded on his merry way to destruction.
Unfortunately for him, his proof involved claiming that the areas of triangles are in the same ratios of their height. Each one of us learned freshman year that they are in the duplicate ratio of their height. However, most of the class wasn't even playing attention by this point, so we didn't notice, but something about the way he said, "You'll grant me that..." bothered me, and I stared at the line for a bit. He was proceeding to the next line when I stopped him by pointing to his line about the direct proportion and stated, "That's wrong."

It felt like there was a collective gasp from the class, followed by surreptitious high-fives and grins.

"Triangles are in a duplicate ratio to their heights," I continued.
He fell deeper into the hole he dug.
"No they're not, they're in duplicate ratio to their sides, and as I wrote, in direct ratio to their heights."
"But in a right triangle.." I postulated... He cut me off, hastening his demise.
"Just look at this triangle," he said, drawing exactly the right triangle I wanted on the board for me. "here's the first triangle, and I'll draw the second on top, with double the side. See? The area is in the ratio of the height."
I leaned from my seat, sketched two lines across his drawing, and quipped, "No. Four times the size, duplicate ratio. It's the most obvious thing possible."
As his face fell, I heard Dr. Decaen muttering causticly to himself behind me, "No, actually it's Ex Aequali that's the most obvious thing possible, as Mr. B said earlier, because even I don't understand it, and I've got a Phd."

"Well, then, I'm not sure what's wrong," wondered the student, "It has to be in a direct ratio, otherwise my proof wouldn't work, and it proved the same thing as Newton's."
"Actually," Mr. Decaen observed, "If you took that up it would prove the opposite of Newton's, violating the law of non-contradiction." (which is a sacred truth at our school, because if you question that...well, you've got some things to work out, shall we say)
"However," I continued, "If you use a duplicate ratio there, that mode of proof would work perfectly, in fact agreeing with Newton completely."

We spent a few minutes discussing how it would work, and which points of the proofs were shown off by the different methods, but it seemed like the wind had left Mr. B's sails, and his brother left the room shortly after. I had actually not known his brother was there, I just knew a visitor had walked in, but didn't know who.

After class, several of my classmates congratulated me, and told me that this was the one day everyone in my section loved me. Except for Mr. B, of course.