Author's Note: Yay! My first Ace Attorney fanfic!

Basically, this was born from my crazy mind after a very slap-happy day in math class. (Speaking of which...That girl excited about proofs tooootally isn't me. Not at allll... ;D) So I hope that all you fellow math-lovers (And even people who, sadly, hate the subject) enjoy this. :D

Also, in case you're having trouble picturing it, the diagram the teacher draws is also the cover image. It got cut off, but the bottom left corner is B, and there's a point on the far right, on the same line, that's F.

I don't own Ace Attorney, and constructive reviews are always welcome. :)

Ms. Polly Nomial, the most easygoing Accelerated Geometry teacher at Themis High School in Los Angeles, watched as her students prepared for the week ahead. She glanced around the room, watching the excited kids looking forward to starting a new unit, the procrastinators finishing their homework at the last minute, and the bored people staring at the wall. Her gaze rested on the formally dressed perfectionist wearing a maroon button down shirt, black dress pants, and a white, ruffly cravat. He was frowning disapprovingly at the boy across the aisle, dressed casually in jeans, a deep blue t-shirt, and red sneakers, and gesturing to the black, mess of hair on his best friend's head. The other gave a small smile and shrugged, as if asking "What do you want me to do about it?" Just as the final bell rang, the gray haired student plucked a brush out of his perfectly clean and organized bag, and handed it to his chaotic friend while slyly smirking. The teacher watched the boy roll his eyes and run the brush through his hair, his trademark spikes taking shape, before deciding to call her brightest, and most boisterous, class of freshmen to order.

"Good morning, everyone!" All but the most eager students responded with the stereotypical, tired grumble of "g'morning". Despite the dismal response, the exuberant teacher continued.

"Today we're going to learn a new type of proof!"

If Ms. N, as her class called her, had thought the students seemed disinterested before, she could be certain as they let out a collective groan, and turned as one to glower at the nerd who had exclaimed "Yes!", complete with a grin and pumped fist. The overly zealous girl gave a sheepish grin and shrunk down in her seat at the annoyance emanating from her peers, especially the owner of the earlier hairbrush, who had seemed to master the art of frigid glares.

Still, Ms. N grinned even more largely and went on. "However, instead of telling you exactly how this works, I want you to figure it out for yourselves." Her students all pulled out notebooks and pencils as she placed a sheet of paper under the ELMO. She drew a diagram of a triangle, with a smaller one inside it, for her students to copy. "The given information is that segment AB is parallel to segment DE, and angle DEC is congruent to angle FCG. With your row partner, prove that triangle ABC is not isosceles." As she put her pen down, she noticed two particular boys had perked up at "partners", and were now smirking at each other across the row. She wondered again if it was a bad idea to seat them together. Nah. She thought, chuckling under her breath. They get so enthusiastic. And besides, it's quite entertaining...

"...ABC is not isosceles." Miles Edgeworth and Phoenix Wright exchanged competitive smirks as Ms. Nomial finished talking.

"I think it's my turn," Phoenix began, "but before I start, here's your hairbrush back." Miles took the object out of his friend's hand, smirk growing.

"Your hair looks much better now, you know."

Wright rolled his eyes, grinning. "Anyway, get ready for some flawless logic!"

Edgeworth's countenance shifted to one of skepticism. "Wright...You really need to work on improving your sense of humor."

"Oh, ha ha. Very witty."

"You should work on your laugh, too."

"Just let me start already, will you?" Phoenix took his friend's silent smile as encouragement to continue. "We are given that segment AB is parallel to segment DE, and angle DEC is congruent to angle FCG. ABC and DEC are corresponding angles, and are thus congruent by the Corresponding Angles Postulate."

"Hold it!" Edgeworth's voice got slightly louder as he interrupted Phoenix by banging on his desk. "That's a theorem, not a postulate."

"Oh, same difference."

"Actually, they're very different."

Phoenix ignored this. "Moving on, since angles DCE and FCG are vertical angles, and vertical angles are congruent, DCE and FCG must be congruent. We can now determine, since DEC and FCG are congruent, and ABC and DEC are congruent, that ABC and FCG are congruent using the...The…"

"The justification you are looking for…" Edgeworth skimmed their textbook, Geometry Records, to find what he needed. "Take That!" His voice was louder still as he shoved the book towards his friend's astounded face, pointing at a specific line. "You need the Transitive Property."

Phoenix grinned sheepishly and rubbed the back of his head. "Ack! I should have known that one!" He sighed. "Anyway, ABC and FCG are congruent, and, for that matter, so are ABC and DCE, because of the Transitive Property. We can then use the Base Angle Converse to declare that sides AB and AC are congruent. Lastly, we can now prove that ABC is isosceles because of the Definition of an Isosceles Triangle!" He ended his proof with a smile and a nod.

"OBJECTION!" Miles was now yelling across the room. Just as he stood up, smirking, and tapped his index finger to his head, every student, and Ms. N, turned to watch them, and all the nearby classrooms closed their doors. He continued on, unaware of the sudden drop in volume. "You have just proven that ABC is isosceles, correct?"

"Yes…" Wright suddenly realized where his friend was going with this, and recoiled in shock.

"Then there is something clearly wrong with your logic, because we have been instructed to prove that ABC is not isosceles!" Edgeworth punctuated this statement with a swooping bow, thoroughly convinced that the other's argument had been crushed.

Argh! Wright was starting to panic, unable to find a mistake in his proof. But suddenly, he heard the voice of his tutor, Mia Fey, through his chaotic thoughts, as if she were giving him advice during one of their countless study sessions. "You just have to think about this in a different way! Don't figure out where you went wrong, figure out why you're right!" Of course! If I can just turn my thinking around… That's it!

"OBJECTION!" The eyes of everyone in the room were drawn to the shouter's ostentatious finger pointing at Edgeworth as the former stood. "There is one way that my logic is correct."

"Well, I suppose if we're contrasting it with Larry's logic than it's certainly more comparatively accurate…" A smirk accompanied the jibe on the intelligence of the pair's next closest friend, in regular level math.

"Ms. N is planning to teach us a new theorem!" Phoenix spoke these words with such gusto, that everyone in the room, save one person, wholeheartedly believed this was the case. Despite the passion behind the phrase, however, Miles simply shrugged his shoulders, and shook his smirking head.

"Perhaps your limited memory has forgotten, but Ms. Nomial stated she was going to teach us '...a new type of proof!'."

Wright's mistake cost his argument serious amounts of credibility. He gasped in shock as he tried to recover. C'mon...I know I'm right...But why?! As he pondered, he realized the mistake in the words his friend had just uttered. Phoenix smiled as he finally understood the answer that had been staring them in the face all along.

"OBJECTION!"

"Wright, if this turns out to be a nonsensical point again...I think you're going to have to forfeit the debate." Edgeworth wagged his finger in the direction of his opponent.

"Don't worry. I've got the answer this time." Phoenix's smile grew as he continued. "The reason that what I've proven contradicts what Ms. N told us, is that such contradictions are necessary for this new type of proof!"

"Hah! I assume you have evidence to back up your conclusion. After all, evidence is everything in a proof!" Despite his words, Miles clearly believed that Phoenix had nothing to back up his statement, which, of course, was the truth. The latter flipped through the textbook, looking for anything that could help substantiate his claim. Finally, he found it: the first section in chapter 11.

"TAKE THAT!" This interjection was the loudest yet, as Phoenix pointed to the introduction paragraph. "Thus far, we have only studied proofs where everything must be proven true. However, there is one type of proof in which we must first prove that one condition is false before being able to assert the true condition. In his chapter we will learn the structure and reasoning behind the indirect proof, an important part of geometry and math as a whole."

The whole room became silent for a few seconds while everyone digested this new information. Then, out of nowhere, Edgeworth pounded his desk with both arms, leaned forward with narrowed eyes, and bellowed "Nnnghhhoooh!"

Ms. Nomial watched as Miles bellowed in defeat. "I think that a clear winner can be determined at this point. I see no point in continuing this debate. Phoenix is correct, and we will be starting the section on indirect proofs tomorrow." The bell rang as she quickly wrapped up her short monologue. "There's no homework tonight. Have a good day, everyone!"

Ms. N smiled as the students gathered their things and rushed out the door. She turned around and drew a single, short line under the column of the T-Chart that represented Phoenix. As she noticed that this new tally brought the score for both of the extremely competitive friends to a tie, she heard a voice behind her.

"But...but logic's MY special power!" Edgeworth dejectedly muttered to himself as he sulked out the door, still frustrated with his defeat. Ms. N smiled one last time, looking forward to tomorrow's best, most exciting hour of the day.