What is Newton's theorem? Simply stated, if a quadrilateral has an inscribed circle, then the four lines connecting the diagonal and the tangent points of the opposite side have common points.

A bicentric quadrilateral refers to a quadrilateral that has both a circumscribed circle and an inscribed circle.

The problem Li Xuan is doing is an expansion of Newton's theorem to prove the common point problem of bicentered quadrilateral lines. Simply put, it is an enhanced version of Newton's theorem.

There is another theorem involving circumscribed circles, called Pascal's theorem, which is a theorem that the competition party is very familiar with.

Because he knew how to combine poles, Li Xuan quickly thought of a way to prove this question. Although this question appeared in the geometry questions of the CMO League, the difficulty was relatively low in the CMO League, so he would not be able to figure it out even if he racked his brains.

.

"The proof is complete, this problem is solved."

Li Xuan put down his pen and felt very comfortable. Because of this learning buff, the pleasure of solving a difficult problem was invisibly amplified several times, and for a moment, he was filled with an unprecedented sense of accomplishment.

Moreover, with the blessing of god-level buffs, my thirst for knowledge is extremely explosive. After the proof of this question is completed, it is not over yet. I really want to trace the root cause.

Li Xuan was thinking, if the double-centered quadrilateral proves that the conclusion is true, then what about hexagons with inscribed circles? What about octagons? Will it also be true?

The more he thought about it, the more he felt it was a possibility. Li Xuan decided to draw a hexagon first to see if his guess was correct.

Turn on the computer and download the software Geometry Sketchpad.

Li Xuan used software to painstakingly draw a hexagonal figure, and connected the diagonal and opposite sides to the inscribed circle. As he expected, the hexagon was indeed established. If a polygon has an inscribed circle,

The diagonal and the tangent point of the opposite side have the same point.

"My guess is right, and indeed it is correct. If a hexagon has an inscribed circle, the line connecting the diagonal of the hexagon and the tangent point of the opposite side will converge at one point..."

Li Xuan's hand holding the mouse trembled slightly, and he felt a little excited, as if he had discovered a new world.

Although the conclusion is simple, hexagons are not as good as quadrilaterals, so it is a bit troublesome to prove.

"The hexagon should prove..."

Li Xuan picked up the pen and began to prove it seriously. The idea was similar, but the amount of calculation increased. After a while, he proved that if a hexagon has an inscribed circle, the diagonal lines and the tangent points of the opposite sides will converge.

At one point.

The hexagon is established.

What about the octagon?

Continuing from one problem to another, from quadrilaterals to 2n polygons, Li Xuan suddenly thought that he might have found a new theorem. If he could discover a new theorem and prove it to be true, the theorem might be named after him.

Li Xuan was not in a hurry to prove it, and decided to see if his predecessors had discovered this rule. He searched online with great interest, but found that this theorem had already existed, and it was Pencelie's theorem.

"Awesome, someone really discovered this theorem, or was it discovered in the 19th century?"

At that moment, Li Xuan not only admired the wisdom of his predecessors, but also felt a sense of frustration that was two hundred years late.

The path of mathematics is so heroic, and all the paths taken are paths that have been taken by predecessors. Sometimes I accidentally discover a wonderful theorem, and after checking it, I find that the predecessors have already discovered it and even solved the proof.

Knowing that it was Peng Serie's theorem, Li Xuan was a little regretful, but he was not entangled. At this time, his thirst for knowledge exploded, and he was even more curious about how to prove it.

"How to prove Peng Serie's theorem?" Li Xuan was surprised. He couldn't find a proof method on the Internet.

"Forget it, I'll try to prove it myself."

Li Xuan picked up the pen and started thinking.

The proof of Peng Serie's theorem may have gone beyond the scope of Li Xuan's knowledge, and the octagon situation confused him. The proof may be to use the method of analytical geometry to establish a coordinate system and violently calculate it through a large number of calculations.

But he was still very excited and wanted to try to see if he could prove it in a simple way.

When it came to the question of whether the quadrilateral extended to 2n polygons, Li Xuan first thought of mathematical induction. He concentrated on it, his calculation ability and thinking speed increased to the extreme, and he began to prove it.

Of course, this theorem seems simple, but in fact it is not that easy to prove. Li Xuan used mathematical induction to prove that the 2n edge formation was established in an hour. Li Xuan also felt that mathematical induction did not work, but he vaguely felt that

I have already grasped some ideas.

Time flies by as I write, and the god-level buff expires in an hour.

In this hour, he did things that he couldn't usually do for several hours. His problem-solving speed increased and his thinking ability was also enhanced.

The only pity is that Peng Serie's theorem was not proved in the end. It was not that Li Xuan was not smart enough. It involved infinity, so the difficulty became higher. Despite this, Li Xuan also had a feeling of endless aftertaste and still wanted to solve the problem.

But at this moment, Li Xuan always felt that something was wrong. He came back to his senses and immediately turned on his phone. When he saw the message from Lingchu on the phone, he wanted to curse:

"Damn, what a cheating system!"

The system calmly replied: [Don’t be impatient. A truly great academic must pay attention to being alone, love, and hair. They are all worries. Only when they are gone can you become stronger. 】

Li Xuan sneered angrily.

Learning is good, but focusing on solitary students is too much. History has not seen many great scientists who died alone. This buff is a bit tricky and cannot be opened casually.

…

…

At home, Xu Lingchu was alone in the room with his mobile phone in a daze.

Xu Lingchu’s penguin messages sent to Li Xuan were not answered, nor were her text messages. It was like nothing happened. She was wondering if Li Xuan didn’t want her anymore?

She is a bit stingy. I heard that if you like someone very much, messages will be replied instantly. She is the same. If she sees Li Xuan's message, messages will be replied immediately, which means she cares about him very much.

But Li Xuan didn't seem to care about her very much. After an hour, he still didn't reply to the message, nor did he confess to her. He suddenly kissed her without even seriously confessing to her.

Of course, maybe Li Xuan just didn't see the phone.

She was thinking wildly, lying on the table, looking at the screen of her mobile phone in boredom, the light reflecting her young and beautiful face.

At this time, Li Xuan suddenly called.

"Hello, Lingchu?"

Xu Lingchu picked up the phone and pursed his lips to listen to Li Xuan's voice, not wanting to speak.

"I'm sorry, something happened just now and my phone was turned off."

"Ah..." Xu Lingchu was startled and asked nervously, "What's the accident? Are you okay?"

"It's okay."

Although Xu Lingchu was curious, seeing that Li Xuan didn't say anything, he didn't ask any more questions. He just puffed his lips and said, "Okay then."

"What are you doing?"

"I'm reading..." Xu Lingchu felt much better when she heard Li Xuan's voice. Li Xuan would call to prove that he cared about her. She couldn't help but smile, picked up a book and opened it pretendingly.

Get up, "What about you?"

"I'm studying hard."

"You seem to be always learning..."

"Yes, how will I support you in the future if you don't study?"

"Xuan, you are so good-looking, you can sell yourself to support me..." Xu Lingchu typed playfully, and by the end, he blushed and smiled, with small pear dimples showing at both ends of his cheeks.

"Selling myself? Selling it to you?"

"...go away!"

Li Xuan stopped teasing her and sent a message with a smile, "Tomorrow is the weekend, come out and play."

"Okay." She smiled happily, having been waiting for Li Xuan's words for a long time.

…

…

The next day was a sunny day. Li Xuan got up early in the morning, dressed up, and greeted Xu Li before going out: "Auntie, I'm going to the park with my classmates to play today, and I won't be back at noon."

Xu Li found out that Li Xuan was in trouble again, stretched out her neck, and watched Li Xuan go downstairs. She was really curious about what Li Xuan was doing.

When he arrived at the park gate, Li Xuan looked at his watch. He had arrived half an hour early.

There was nothing going on at this time. I was standing in a conspicuous place, looking at the bright sunshine. Thinking of her, the air in front of me had a transparent texture. I was waiting in boredom like this, but suddenly my vision went dark and my eyes were covered by my hands.

"Guess who I am?" A playful voice.

There was no need to guess, but Li Xuan heard her voice, laughed, and looked back.

Xu Lingchu took a step back, staring at him with her hands behind her back and smiling. She was just as beautiful today, with soft air bangs covering her snow-white forehead, pure and lovely, in a light-colored dress, with the skirt reaching to her knees, with faintly visible curves, simple yet elegant.

Both of them came very early. The book says that how far in advance you go on a date means you care more about the other person.

But Li Xuan suddenly realized something was wrong. A black shadow appeared behind her. It was a husky. She was immediately startled: "Er ha??"

The husky with a fierce look on his face looked at him with a frown. He looked at the husky with sharp eyes, and there was no doubt that he was a purebred husky.

Xu Lingchu held the dog leash and blinked.

Seeing Li Xuan's confused look, she couldn't help laughing. Her clear eyes were full of playfulness, her eyebrows were curved, and her smile was like a flower: "Well... I was afraid that you would bully me, so I brought my dog ​​Kaka here...

"

Li Xuan: "..."

Xu Lingchu patted Erha on the head: "My dog ​​is very well-behaved and never bites people."
Chapter completed!