I really just want to be a scholar

Chapter 36 Learning God's Amplification Move, Construction Method!

Cai Jiansen refused to believe it, and hurriedly lowered his head to watch Qin Ke's proof process carefully.

Seeing Qin Kezuo's auxiliary line, Cai Jiansen was relieved, and at the same time was ecstatic, this guy did something wrong!This is different from the standard answer in his hand!
Cai Jiansen almost burst out laughing, no wonder this kid's proof process is less than twenty lines, it turns out he did something wrong!
The auxiliary line drawn in the first step is wrong!
Actually take the midpoint E of AB and the midpoint F of CD as auxiliary lines, respectively connect FN, FE, FQ, FO, FM, then connect EN, EF, EO, EM, EQ, and also connect DQ, DB, CA, AQ, CQ, it's just...it's just...a mess, a mess, this is the price of your arrogance!He actually didn't use scratch paper, just scribbled on the paper!

Hey, wait a minute...

Although the auxiliary lines are drawn more and more complicated, they seem to make sense, not like scribbles.

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

"Proof: Since ⊙O1 and ⊙O2 are equal circles and inferior arcs AQ and BQ subtend the circumferential angles are both ∠BPQ, it can be concluded that AQ=BQ.

In the same way, QC=QD can be obtained, and because the circular angle ∠PAQ=∠PDQ subtended by the inferior arc PQ, it can be obtained
△BQA is similar to △CQD, deduce ∠AQB=∠CQD
……

From this it is deduced that AC=BD,

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

Cai Jiansen's face became darker the more he looked at it, because he found that the method used by this kid was very unusual. He changed it to prove that point O is on the 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...is actually the "construction method" in 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 point of view 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 makes the implicit relationship and nature of the original problem clearly displayed in the newly constructed mathematical object, and the method of solving 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, constructing the relationship in the problem-setting conditions, or imagining these relationships to be realized on a certain model, or constructing the problem-setting conditions through appropriate logical organization a new form.

The construction method has gone through the "intuitive mathematics stage" of Kronick in Germany, Markov's "algorithmic mathematics stage", and then entered the "modern structural mathematics stage" of Bishop, and has been popularized and used in the high school stage mainly in the Olympiad China shines.

But there are not many high school students and even mathematics teachers who are really proficient and flexible in mastering this "construction method".

Because solving problems with the construction method 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 integrate algebra, trigonometry, geometry, number theory and other knowledge from one side Or multi-faceted mutual penetration and organic combination.

However, Cai Jiansen saw such a high school student who has mastered the "structural method" at this time!

Although Qin Ke's proof process only uses the construction method between geometric knowledge points, it also uses the essence of the construction method to the fullest, pointing directly to the core of the proof, simplifying the proof process, and reducing the original need for a whole page. The proof process has turned into a proof process that is less than twenty lines!

Cai Jiansen asked himself that he could not reach such a level in the "structural method"!
This... how high is this kid's mathematical talent!

Cai Jiansen was dumbfounded to see that Qin Ke had completed the second question neatly. Under the agitation of his mind, he only felt that the blood was rushing straight to his brain, and he was a little dizzy because he usually had a little high blood pressure.

He hurriedly took three deep breaths before he could barely calm down his blood, but under the turmoil of his mind, he didn't even notice that Lao Zheng and Vice President Wen came quietly. Soon after, even the rest of the math teachers came. up.

A group of teachers stood quietly behind Qin Ke, watching him answer the questions in shock.

Ning Qingyun noticed something strange, looked back, and couldn't help being shocked.

Old Zheng made a silent gesture towards her, Ning Qingyun nodded in confusion, and followed the eyes of the teachers, only to see that Qin Ke, who was at the same table, was already working on the third question.

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

This... This guy's problem-solving speed is too fast!
He just barely made the first question, but he is already working on the third question?

The girl couldn't help but think of the last midterm exam, Qin Ke only spent 10 minutes to complete all the questions...

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

Ning Qingyun couldn't believe it, but seeing Qin Ke's focused, excited and even confident expression, it was obvious that he was not writing nonsense, but had really finished the first two questions!

This... what kind of monster is this guy!
Qin Ke was completely immersed in the joy of solving the problem and couldn't extricate himself. He didn't notice Ning Qingyun's adoring gaze in shock, let alone a group of people standing behind him.

At this time, he is like a chef who has mastered superb cooking techniques and countless recipes. When he sees a lot of luxurious ingredients, he is full of thoughts on how to make the most satisfying and delicious dishes.

At this time, all he had in his eyes was the Mathematical Olympiad questions that could arouse his strong fighting spirit, and he didn't even hear the system's continuous "Ding! The host has gained double 38 shock points! The experience value of learning the gods + 38" and other consecutive messages. A series of beeps, naturally I didn't see my god of learning experience value bouncing upwards.

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

"3. To prove: for each 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 genuinely difficult problem in the provincial competition!

After thinking about it, Qin Kelue figured out a very complicated process of solving the problem.

He was about to write it down, but a flash of inspiration flashed in his mind, and he thought of a more convenient proof method!
But whether it works or not has to be verified carefully.

Instead of using scratch paper, he closed his eyes and performed mental calculations directly in his brain.

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

More and more students noticed the weird scene here, and couldn't help but look this way, and some even opened their mouths to ask something, but they were stared at by the teachers with sharp eyes, and immediately couldn't help but frightened. Dare to speak out.

The entire classroom fell into an eerie silence, only the sound of breathing of varying severity. Most of the students were not in the mood to do the questions, and looked at this side with puzzled and curious eyes.

(End of this chapter)