Phương pháp chứng minh tứ giác nội tiếp - Toán lớp 9 - Thầy Hồng Trí Quang -HOCMAI

Hello everyone! Today we will start with the form, that is, the proof of the direct form. I see that the subject of any province, any school, all have this part. Without the first sentence, it is the second sentence. And the exercises that have difficult sentences are all related to the direct form of the 9th grade. But in the form of the direct form, I see that many students learn normally. If you don't do the direct form, you will see that you say it's good but you don't understand. It's good but you don't understand. It's really hard to understand. Calculation itself is very difficult. Some people think that even if you have a brain, you can't solve it. That's right. Because normally I don't need such a high level. So I summarized the form to help you take the 10th grade normally. If you set a goal of 8.9, you will see that this part will help you a lot. We will go into 3 methods. I don't divide it much. I divide it into 3 methods. The first is to sum up the two opposite angles. The second is the two angles look the same. The third is to use the formulas in the circle. The other is to analyze to prove the direct form. The first method is to prove the sum of the two opposite angles by 180 degrees. This is the definition of the direct form. The content of the method is that if the ABCD form has a total of two opposite angles of 180 degrees, then that form is direct. In a form, there are two opposite angles. Here I can take this angle A plus this angle C by 180 degrees. Or this angle B plus this angle D by 180 degrees. One of the two ways to prove that is to conclude the direct form of the ABCD form. Direct form means that there are 4 points on a circle. Now I will go to some results. The first result is for the ABCD form. If this angle A is square and this angle C is also square, then the ABCD form is the direct form. Because the sum of these two angles is equal to 180 degrees. And because these two angles are square, this form has something special. That is the direct form of the circle line, the BD line. And this center O is the center point of BD. Why do I have this? Remember that if this angle is square, then the center point of BD is the center point of AO by half of BD. That is OA by OB by OD. Similarly, if the angle C is square, then OC by OD by OB. Therefore, OA by OB by OC by OD, then 4 points of ABCD are on the circle line O. Next, because the sum of these two angles is also equal to 180 degrees, so to prove that this angle plus this angle is equal to 180 degrees, we will prove that these two angles are equal. So we have the second result. Secondly, if the angles E, A, D are equal to the angles B, C, D, then we also get the direct form. So we have two special effects of the first method, which is to prove the sum of two opposite angles. Now, before going into the exercise, I would like to remind you of the knowledge that is related. First, the two opposite angles are equal to each other, and they are equal to half of the sum of the opposite angles. If you forget, remember now. It's okay if you forget, remember now. For example, if you read for me which two angles are equal to each other, then I have the angle ADB by the angle ACB, which is the two opposite angles are equal to each other. Next, what is the relationship between the opposite angle and the center angle? For example, the opposite angle AB, the opposite angle is the angle ADB, and the center angle is the angle AOB, then the opposite angle is equal to half of the center angle, which is equal to half of the sum of the opposite angles. Because the measure of the center angle AOB is equal to the measure of the center angle AB, and the measure of the center angle is equal to half of the sum of the opposite angles, so the angle D is equal to half of the sum of the opposite angles. If you are afraid of shapes, if you are still weak in shapes, it's okay. We have finished the sum of the teacher here, about 3-4 exercises. If we continue, we will be able to improve quite a bit. If you want to improve more, we will continue to do the part of proving that the same rule is true for all formulas, 5 more formulas, we will see that the shape is very good. Next is the origin of the convection and the arc. Equal to the inner angle, but without an arc. Now I draw the convection. The convection AX is the angle with the radius OA at point A, so AX is the convection. The origin of the convection and the arc AC is the angle CAX. CAX is equal to the inner angle, but without an arc. Here is the arc AC. So here is the inner angle, which is the angle D. These two angles are equal. And according to the above equation, the angle COA is the center angle, so it will be twice the inner angle, but without an arc. So COA is twice the angle D. And this angle D is equal to this angle A. So the angle COA is twice the angle CAX. And now we will go to the exercise. Do you remember all the formulas? Remember, right? It's simple. The convection angle is equal to the inner angle. The convection angle is equal to the center angle. The origin of the convection and the arc is equal to the inner angle, but without an arc. There are 3 cases like that. Lesson 1, this exercise is about starting. It's very simple. But remember the special nature of the mirror. For the circle O, the angle AB. And there is also an angle MN that is not square with AB. Not square, just to make sure we draw it. It doesn't matter. Proceed at B, cut AM at C and cut AN at D. Prove that the square CMND is the inner square. You need to learn the inverse order. What does Debye want to prove? Let's turn it upside down. To get this, you need this. This is the fastest and smartest way to learn. For example, to prove the square CMND is the inner square, we need to prove that these two angles are equal. According to the equation 1, we will continue. These two angles are equal. We will get this angle plus this angle is equal to 180 degrees. How to prove these two angles are equal? There are many ways. First, we need to find the center angle. This angle C is equal to that angle and that angle is equal to that angle. Second, we need to prove that this angle is equal to another angle. This angle is also equal to that angle. If these two angles are equal to each other, then they are equal. Or if they are not equal, then we will see where they meet. Before we find out how to prove these two angles are equal, I will go and explore more. The circle AB is connected to B. So the angle is square. Next is the circle MN. So this angle is also square. All the angles that look at MN are square. MBN is also square. Circle AB. So AB is square with CB. And the angle AMB is also square. There are many ways to prove that angle N is equal to angle C. The center is also equal to that angle. And we can also prove that it is equal to that angle. Now I will take an example. I will try to prove that angle is equal to that angle. There are many ways. This angle is equal to this angle. Let's remember. A triangle. This top angle is equal to this bottom angle. Equal to this top angle. Why? Because these two angles are equal to the angle A. And this angle is equal to this angle. Because the triangle AON is a triangle at O. So I will prove that the center is this angle. Now let's continue to prove. Let's start from the beginning. Let's prove the following. The first is to prove that the angle N is equal to the angle A. Because the triangle OAN is a triangle at O. So I will prove that the angle OAN is equal to the angle ONA. Why is OAN equal to the angle ACB? I will argue that these two angles are square. Because MN is a line. So the angle CAD is equal to 90 degrees. Because CD is a line at B. So the angle AB is equal to the angle CD. So this angle plus this angle is equal to 90 degrees. This angle plus this angle is equal to 90 degrees. So these two angles are equal. Because they are equal to the angle CAB. From the above two lines, we get the angle ACB. ACB is equal to the angle BAN. Equal to the angle CAB. So this angle is the middle angle. B OAN. These two angles are equal. So this angle is equal to this angle. Equal to the angle ACB. ACB is equal to the angle ONA. So we get the total of two angles. The angle ACD plus the angle MND is equal to 180 degrees. Equal to the square. The square MNDC is the inner square. As shown in the figure. So we have to prove it. So the first exercise is quite simple. It's just about starting. Let's move on to the second exercise. Let's continue with the second exercise. Put the inner square ABC in the circle. The height AH. The inner square in the circle is a bit common. Just put the inner square ABC. The height AH. The height HE is the square with AB. The height HF is the square with AC. Let's prove it. The inner square AEHF is the inner square. The inner square BEFC is also the inner square. Remember that this angle is the square angle. The first exercise is quite simple. The inner square AEHF is the first result. These two angles are the square angles. We can prove that this square is the inner square. We even know that it's on the circle line AH. The first exercise is done. The second exercise is BEFC, the inner square. BEFC We have to exclude the way to prove that these two angles are the same angle. Although you have learned it, please note that we will not go in that direction. Why not? Don't tell the teacher that you are learning about the opposite angle. You have to think about it. We have three methods, right? We have to use a faster method. There is no slope here. There is no BEFC, there is no AC, there is no EEC. So we limit the creation of more images. We use images with slope. We will think about proving that this angle plus this angle is 180 degrees. This angle is 90 degrees. We will prove that this small angle plus this angle is 90 degrees. That's one way. Or we prove that this angle is equal to this angle according to the result we need to prove. Either way is fine. Now I just need to know what to prove. I will have a direction to prove. For example, if you want to prove that this angle plus this angle is 90 degrees, we have a comment that this angle plus this angle is 90 degrees. If you prove that these two angles are the same, it's done. Of course they are the same. If you want to prove that this angle plus this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle plus this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 90 degrees. If you want to prove that this angle is 90 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. If you want to prove that this angle is 180 degrees, you have to prove that this angle is 180 degrees. You have to prove that this angle is 180 degrees. Some of you may think that this is the girl of your life. There is nothing for you to know that, right? You just have to feel it. And the first property that we know If Bi and Ci are fractions Then Ai is also a fraction familiar but not proven Or there is proof but you forget These two are the same proof But because I I haven't proved the first plus sign yet So I don't know how to prove the second plus sign So it's very simple I prove this If you want to prove Ai is a fraction Then we draw two squares here And we prove that I is equal to both sides For example, drawing AH and AK is a square to AB and AC Then we prove that IH is equal to IK If there is IH is equal to IK Then we subtract AIH by AIK And so these two are equal Then AI will be a fraction In short, we remember one thing The point Y Which is on the fraction line When and only when the distance from Y to these two sides is equal If there is a fraction Then this distance is equal Because these two sides are equal And if the two square distances are equal Then these two sides are equal And the point is a fraction So now I prove The first proof is that if VI and CI are fractions Then AI is a fraction By subtracting 3 square lines down here First is because If CI is a fraction Then these two distances are equal Next is BI is a fraction Then these two distances are equal So these two distances are equal Because they are equal to the distance below If these two are equal, then AI is a fraction And this proof With the proof that AI is a fraction is exactly the same From this E, you also subtract 3 square lines And you prove it that way However, this property I was recognized Back to this property This property is in class 7 We have learned Speak as follows Two fractions of two square angles are square with each other Two fractions of two square angles are square with each other However, I still have to prove For the calculation Now let's move on to the B argument Call M the midpoint of E M is the midpoint of E Prove that M is the outer circle of ABC Prove that M is the outer circle of ABC This way of speaking Equals to proving that ABMC is an inner square Now I think M is the midpoint of E But this sentence We have just proven that The inner circle of BECE The inner circle of BCE