Leah once said, "the success of solving a problem depends on the choice of correct thinking and on attacking the fortress from a direction that can be close to it." In order to distinguish which way of thinking is correct and which direction can be approached, we need to explore various directions and ideas.
Road. It can be seen that mastering the general idea of proving problems, exploring mathematical thinking in the process of proving problems and summarizing the basic laws of proving problems are the keys to solving geometric proving problems. The common ways of thinking to prove problems are direct thinking and indirect thinking.
Road.
First, think directly.
When proving a problem, we should first carefully examine the meaning of the problem, carefully observe the problem, distinguish the conditions from the conclusions, and try our best to dig out some problem-solving information hidden in the problem, so as to make a series according to definitions, formulas and theorems on the basis of careful examination of the problem.
Logical reasoning, and finally get the proof of the proposition, this is called direct thinking. Because of the backward way of thinking, the main methods to prove the problem are "analytical method" and "comprehensive method"
1. Analysis method. The analysis method is to start with the conclusion of the proposition, first admit that it is correct, and then find the conditions for the correct conclusion, step by step, until it meets the topic, so the topic is established.
The thinking process of drawing a conclusion. In the process of tracing the source from the conclusion to the known conditions, the forms of tracing the source will be different because of the different conditions of the topic or the different implicit degree of the relationship between the known conditions, so
Analysis methods are usually divided into the following four categories.
(1) selectivity analysis method. To solve a problem with selective analysis, we should first start from the conclusion A of the topic, and gradually turn the problem into an analysis to get what sufficient conditions are needed for conclusion A. Assuming that there is condition B, there will be a conclusion.
A, then B becomes a sufficient condition for choosing to find A, and then analyzes under what conditions B can be chosen ... and finally traces back to a certain condition in the proposition.
(2) Reversible analysis method. If every step is a necessary and sufficient condition in the process from the conclusion to the known condition, then this analysis method is also called reversible analysis method, so the reversible analysis method is selective.
A special case of analytical method. The proposition proved by reversible analysis can be proved by selective analysis, but the proposition proved by selective analysis cannot be proved by reversible analysis.
(3) Structural analysis. If in the process of tracing back from the conclusion to the known conditions, we need to take corresponding structural measures in the "fork" of finding new sufficient conditions, such as constructing some conditions and making some auxiliary diagrams. Then we can trace back to the known conditions of the original proposition, which is called structural analysis.
(4) Scenario analysis. In the process of tracing back to the known conditions, with the help of well-founded assumptions and assumptions, a new "plausible" idea is formed, and then the "well-founded" verification is carried out to gradually find out the correct way, which is called hypothesis analysis.
2. Comprehensive method. The synthesis method starts from the conditions of proposition, leads from cause to effect, approaches the goal step by step through a series of correct reasoning, and finally draws a conclusion. Starting from the known conditions and according to the known definitions, formulas and theorems, the conclusion is gradually deduced. In this process, due to different thinking angles and positions, the comprehensive method is often divided into four types:
(1) Analytical synthesis method. We turn the narrative of analytical method to solve problems in reverse, and the solution obtained by a little sorting is called analytical synthesis method.
(2) A comprehensive method based on foundation. When it is difficult to start with known conditions, or there is no familiar model for induction and deduction, we can turn to a simple model and then reduce the general situation to this simple model. This synthesis method is called the basic synthesis method.
(3) Mediated synthesis. When the given conditions of the problem are few and not directly related to the conclusion, or when the conditions are loose and difficult to use, we should consciously find, select and use the media to realize the transition. Such a comprehensive method is called media-based comprehensive method.
(4) Analytical synthesis method. When solving the problem, the overall scheme and direction of solving the problem are formulated by using the idea of analytical method, and then the scheme is not really realized by analytical method, but by comprehensive method, which is called analytical synthesis method.
These two methods can be used separately or in combination in concrete proof, with synthesis in analysis and analysis in synthesis for cross-use.
Second, indirect thinking.
Some propositions are often difficult or even impossible to prove directly. At this time, we might as well prove its equivalent proposition to achieve our goal indirectly. This way of proving is called indirect thinking. We often use the reduction to absurdity and the same proof to prove problems, which are two typical methods to prove problems with indirect thinking.
1. Reduce to absurdity. Specifically, when proving a proposition, if it is not easy to start from the front, we should start from the opposite side of the proposition conclusion and assume that the opposite side of the conclusion is established first. If strict reasoning is carried out based on this assumption, the derived result is the same as the previous one.
One of the known conditions, formulas, theorems, definitions, assumptions, etc. It is contradictory, or two contradictory results are derived, which proves that the assumption that the opposite side of the conclusion is true is wrong, so that the positive side of the conclusion is true.
This method of proof is called reduction to absurdity. When the conclusion has only one negative side, denying this one completes the proof. This simple reduction to absurdity is also called reduction to absurdity When the conclusion has several opposing aspects, it must be refuted.
Each kind, this tedious reduction to absurdity is also called exhaustive method.
There are usually three steps to prove a problem by reducing to absurdity:
(1) reverse design. Making assumptions contrary to the conclusion is usually called disproof hypothesis.
(2) Return to absurdity. By using assumptions that are contrary to the known conditions, logical reasoning is carried out, and the results that contradict the known conditions, axioms, definitions, etc. are obtained. Are all derived. According to the law of contradiction, in the process of reasoning, we can't make two opposite judgments on the same object at the same time and in the same relationship, which shows that the opposite assumption is untenable.
(3) draw a conclusion. According to the exclusion rate, that is, in the same demonstration process, proposition C and proposition non-C have and only one is correct, we can know that the original conclusion is true.
2. The same law. If you want to prove that a graph has a certain property, but it is complicated or difficult to prove it directly, sometimes you can make a graph with the indicated property, and then prove that the graph made is the same as the given graph, so as to make them equal. This method of proof is called the same method.
For example, the steps to prove a plane geometry problem by the same method are: (1) making a graph that conforms to the proposition conclusion; Prove that the graph meets the known conditions; According to the uniqueness, it is determined that the produced graphics are consistent with the known graphics; Determine the truth value of the proposition.
The same method and reduction to absurdity are both indirect ways of thinking. Among them, the same method has great limitations and is usually only applicable to propositions that conform to the same principle; The application scope of reduction to absurdity is wider. Propositions that can be proved by reduction to absurdity may not necessarily be proved by the same method, but propositions that can be proved by the same method can generally be proved by reduction to absurdity.
In the process of proving a problem, whether it is direct or indirect, a series of correct reasoning should be carried out, which requires the problem solver to analyze, process and transform the confusing appearance from the outside to the inside, discard the false and retain the true, explore from different directions, choose ideas in a wide range, correct the mistakes in the attempt in time, and finally obtain the proof of the proposition.