Cai Jiansen refused to believe it and hurriedly lowered his head to take a closer look at Qin Ke's proof process.

As soon as he saw the auxiliary line made by Qin Ke, Cai Jiansen was relieved and at the same time he was ecstatic. This guy made a mistake! This was different from the standard answer in his hand!

Cai Jiansen almost laughed out loud. No wonder this guy's proof process was less than twenty lines. It turned out that he did it wrong!

The auxiliary lines drawn in the first step were wrong!

Actually, we took the midpoint E of AB and the midpoint F of CD as auxiliary lines and connected them respectively to FN, FE, FQ, FO, FM, then to EN, EF, EO, EM, EQ, and also to DQ and DB.

CA, AQ, CQ, it's just... it's just... a mess. This is the price of your arrogance! You don't even need scratch paper, just scribble directly on the paper!

Hey, wait a minute...

Although the auxiliary lines are drawn more and more complicated, they seem to make sense and don't look like graffiti.

Cai Jiansen couldn't help but look at the proof process written by this guy:

"Proof: Since ⊙O1 and ⊙O2 are equal circles and minor arcs AQ, and the circumferential angles subtended by BQ are all ∠BPQ, it can be concluded that AQ=BQ.

In the same way, we can get QC=QD, and because the circumferential angle ∠PAQ=∠PDQ subtended by the minor arc PQ, we can get

△BQA is similar to △CQD, and it is deduced that ∠AQB=∠CQD

…

From this it follows that AC=BD,

It can be concluded that NEMF is a rhombus, and it can be deduced that M and N are on the mid-perpendicular line of EF..."

The more Cai Jiansen looked at it, the darker his face became, because he found that the method used by this boy was very unusual. He changed it to prove that point O is on the mid-perpendicular line of EF, thus proving that the three points M, N, and O are collinear!

It is actually simpler and easier to understand than the proof method he made!

What this kid used... was actually the "Construction Method" from the Mathematical Olympiad!

Cai Jiansen was completely stunned.

"Construction method" is a very important problem-solving thinking in Mathematical Olympiad.

It refers to observing, analyzing, and understanding the object from a new perspective based on the characteristics and properties of the conditions and conclusions of the question, and then using known mathematical relations and theories as tools to construct in thinking that satisfies the conditions or

The mathematical object of the conclusion enables the relationships and properties implicit in the original problem to be clearly displayed in the newly constructed mathematical object, and it is a method to solve mathematical problems conveniently and quickly with the help of this mathematical object.

Generally, in the actual problem-solving process, there are three main construction methods. Construct the relationships in the problem conditions, or imagine these relationships to be realized on a certain model, or construct the problem conditions through appropriate logical organization.

come up with a new form.

The construction method has gone through the "intuitive mathematics stage" of German Kronich and Markov's "algorithmic mathematics stage" before entering Bishop's "modern construction mathematics stage", and has been promoted and used in high school.

Zhongda shines.

However, there are not many high school students and even mathematics teachers who are truly proficient and flexibly master this "construction method".

Because the constructive method of problem solving has extremely high requirements on students' mathematical talents, it requires students to have extremely comprehensive knowledge and keen intuition, to be able to associate from multiple angles and channels, and to combine algebra, trigonometry, geometry, number theory and other knowledge from one aspect.

Or multiple aspects can penetrate each other and combine organically.

It just happened that Cai Jiansen saw such a high school student who used the "Construction Method" proficiently at this time!

Even though Qin Ke's proof process only uses the construction method between geometric knowledge points, it also fully utilizes the essence of the construction method, directly pointing to the core of the proof, simplifying the proof process, and reducing the original need for a whole page.

The proof process is reduced to a proof process that takes less than twenty lines!

Cai Jiansen asked himself that he could not reach such a level in "structural method"!

This... this kid's mathematical talent is so high!

Cai Jiansen was dumbfounded to see Qin Ke finish the second question neatly. He was so excited that he felt his blood surge and rush to his brain. He, who usually has high blood pressure, suddenly felt dizzy.

He took three deep breaths and managed to calm down his energy and blood. However, in his mental turmoil, he didn't even notice Lao Zheng. Vice-principal Wen also came quietly. Soon after, even the other math teachers came as well.

.

A group of teachers stood quietly behind Qin Ke, watching him answer questions with shock on their faces.

Ning Qingyun noticed something strange, looked back, and was startled.

Lao Zheng made a silent gesture towards her, and Ning Qingyun nodded in confusion. She followed the teachers' gazes and saw that Qin Ke, who was at the same table, was already doing the third question.

The girl's beautiful eyes widened instantly, revealing an unconcealable shock.

This... this guy's problem-solving speed is too fast!

He had just managed to solve the first question, and he was already doing the third question?

The girl couldn't help but think of the last midterm exam. Qin Ke only spent twenty minutes to complete all the questions...

But this paper is not an ordinary test question, but a big question that is close to the difficulty of the provincial semi-finals!

Ning Qingyun couldn't believe it, but looking at Qin Ke's focused, excited and even confident look, it was obvious that he was not writing blindly, but had actually finished the first two questions!

This... what kind of monster is this guy!

Qin Ke was completely immersed in the joy of solving problems and couldn't help himself. He didn't even notice Ning Qingyun's shocked eyes with faint admiration, nor did he know that there were a bunch of people standing behind him.

At this time, he was like a chef who had mastered superb cooking skills and countless recipes. When he saw the piles of luxurious ingredients, he was thinking about how to make the most satisfying and delicious dishes.

At this time, he could only see the Mathematical Olympiad questions that aroused his strong fighting spirit, and he did not even hear a series of messages such as "Ding! The host has received double the shock value of 38 points! The God of Learning experience value is 38" constantly coming from the system.

There was a notification sound, and naturally I didn't see my God of Learning experience points jumping upwards.

After proving the second question, Qin Kema looked non-stop at the third question, which was also a proof question.

"3. Verify: For every positive integer m, there is a finite non-empty point set S in the plane, which has the following properties: for any point A∈S, there are exactly m points in S with a distance of 1 from point A.

"

It’s a bit interesting, this is a truly difficult question in the provincial competition!

After Qin Ke thought about it for a while, he mentally calculated a very complicated problem-solving process.

He was about to write it down, but an idea flashed in his mind and he thought of a more convenient way to prove it!

But whether it works or not needs to be verified carefully.

He did not use scratch paper, but closed his eyes and performed mental calculations directly in his brain.

Seeing Qin Ke closing his eyes and meditating like an old monk in trance, the teachers subconsciously held their breath and did not dare to disturb his train of thought.

More and more students noticed the strange scene here, and couldn't help but look over here. Some even opened their mouths to ask something, but were stared at by a group of teachers with sharp eyes, and they were so frightened that they immediately stopped.

Dare to speak out.
Chapter completed!