On the sine theorem, In the triangle ABC, if a / cosa = B / CoSb, then the shape of the triangle is________ . (isosceles triangle, I can understand why it turns to the end, Tana = tanb, and why can't a + B be equal to 180?)

Because in a triangle, if a + B = 180 degree, then C = 0 degree, it will not form a triangle
So a + B can't be equal to 180 degrees

Sine theorem
In the triangle ABC, a: B: C = 4:1:1, then a: B: C equals ()

4;1;1

What are the criteria for four points to be concentric?

At 15:44 on August 4, four points are in a circle: first of all, the four points are on the same plane. If you can find a circle on the plane and make the circle pass through the four points, it can be called the four points are in a circle
Professional point is: four points on the same plane, if there is a circle through the four points, then it is called four points in a circle
If you think that any two points on a circle are connected to form a string, the vertical bisector of the string must pass through the center of the circle
A. The four points B, C and D are on the same plane. If the vertical bisectors of AB, BC and CD intersect at one point, then the four points are in a circle, and the intersection point is the center of the circle
It is proved that: if the intersection point is O, then o is on the vertical bisector of AB, BC and CD. According to the distance from the point on the vertical bisector to the two ends of the line, there is OA = ob = OC = OD. Then the circle with o as the center and OA as the radius must get a circle through a, B, C and D. these four points are in common circle
The reason why we want to study the four points co circle is that the three points must be co circle. You can prove it with the above idea, but we also need to use "the vertical bisectors of the three sides of the triangle intersect at one point". The center of the circle obtained here is the "outer center"

What is the condition that four points are in the same circle in geometry?

There are two cases
First, choose any two points as the center vertical line, and the other two points as the center vertical line. If the intersection of the two center vertical lines is equal to the distance from four points, then four points are in a circle
Second, if there are two right triangles with four points as the hypotenuse points of the two right triangles, then the four points are in a circle

What are the conditions for four points to be round
Is there a matrix expression without equation solution?

Determination of the perpendicularity of a straight line plane
Questions like this
Given a (2,0), B (3,5), the line L passes through point B and intersects with y axis at point C (0, y). If oabc four points are in a circle, then the value of Y is
What ideas should we start from and how to do to solve such examples and detailed solving process?
Note that this kind of question is not just to solve this problem, you can paste examples!

Let's take your problem as an example. You first draw a coordinate axis, mark several coordinates, and then draw a circle to make these points on it. It is a quadrilateral with a right angle. If you connect Ca, there will be two circumference angles, one is the angle COA, the other is the angle CBA. They add up to 180 degrees, so the angle CBA

What are the criteria for four points to be a common circle?
Can a group of circle angles be equal to prove that four points are round? I was told that there should be at least two groups

A pair of opening angles are equal, or a pair of inner angles complement each other

What's the condition of four points in the plane being co circular?

The diagonal sum of four points is 180 degrees
If you don't understand, please hi me, I wish you a happy study!

What is the necessary and sufficient condition for four points to be concentric

There are some basic methods to prove that four points are common circles
Method 1: select three points from the four points to make a circle, and then prove that the other point is also on the circle. If we can prove this point, we can confirm that the four points are all round
In method 2, four points proved to be co circular are connected into two triangles with common base, and the two triangles are on the same side of the base. If it can be proved that their vertex angles are equal, then it can be confirmed that the four points are co circular. (if it can be proved that their two vertex angles are right angles, then it can be confirmed that the four points are co circular, and the line between two points on the oblique side is the diameter of the circle.)
In method 3, four points proved to be cocircular are connected into a quadrilateral. If it can be proved that they are diagonally complementary or that one of their outer angles is equal to the inner diagonal of their adjacent complementary angles, then the four points are sure to be cocircular
Method 4 connect the four points of the proved common circle into two intersecting line segments. If it can be proved that the product of the two line segments divided by the intersection points is equal, the four points of the proved common circle can be confirmed; or connect the four points of the proved common circle and extend the intersecting line segments, If we can prove that the product of two line segments from the intersection point to the two ends of a line segment is equal to the product of two line segments from the intersection point to the two ends of another line segment, we can confirm that the four points are also co circular. (according to the inverse theorem of Ptolemy's theorem)
Methods 5 prove that the distance from the point of common circle to a certain point is equal
The basis of each of the above five basic methods is one of the reasons why four points are in common circle. Therefore, when we want to prove the problem of four points in common circle, we should first choose one of the six basic methods according to the conditions of proposition and the characteristics of graphics
Judgment and nature:
The diagonal sum of inscribed quadrilateral is 180 degrees, and any outer angle is equal to its inner diagonal
If the quadrilateral ABCD is inscribed in circle O, extending AB and DC to e, crossing point E as tangent EF of circle O, AC and BD to P, then a + C = 180 degrees, B + D = 180 degrees,
Angle ABC = angle ADC (equal to the circumferential angle opposite the arc)
Angle CBE = angle D (outer angle equals inner angle)
Δ ABP ∽ DCP (three internal angles are equal)
AP * CP = BP * DP (intersection chord theorem)
The picture EB * EA = EC * ed (secant theorem)
EF * EF = EB * EA = EC * ed (cutting line theorem)
(cutting line theorem, secant line theorem and intersecting string theorem are collectively referred to as circular screen theorem)
AB * CD + ad * CB = AC * BD (Ptolemy's theorem)

What is the condition for four points to be in a circle?

Proof
There are some basic methods as follows:
Method 1: select three points from the four points that are proved to be a circle, and then prove that the other point is also on the circle. If we can prove this point, we can confirm this point
．
Method 2 connect the four points of the proved common circle into two triangles with the same base, and the two triangles are on the same side of the base. If we can prove that their vertex angles are equal, we can confirm the result
If it can be proved that the two vertex angles are right angles, it can be confirmed that the four points are in a circle, and the line between the two points on the hypotenuse is the diameter of the circle
In method 3, four points proved to be a common circle are connected into a quadrilateral, if it can be proved that they are diagonally complementary or one of them can be proved
Equal to its
Of
The four points are sure to be round
Method 4 connect the four points of the proved common circle into two intersecting line segments. If it can be proved that the product of the two line segments divided by the intersection points is equal, the four points of the proved common circle can be confirmed; or connect the four points of the proved common circle and extend the intersecting line segments
The product of two line segments from a point to two ends of a line segment is equal to
The product of two line segments from a point to the two ends of another line segment confirms that the four points are also in a circle
Of
)
Methods 5 prove that the distance from the point of common circle to a certain point is equal
The basis of each of the above five basic methods is one of the reasons why four points are in common circle. Therefore, when we want to prove the problem of four points in common circle, we should first choose one of the six basic methods according to the conditions of proposition and the characteristics of graphics
Judgment and nature:
The diagonal sum of is 180 degrees, and any one
It's all the same
.
If the quadrilateral ABCD is inscribed in circle O, extending AB and DC to e, crossing point E as tangent EF of circle O, AC and BD to P, then a + C = 180 degrees, B + D = 180 degrees,
Angle ABC = angle ADC
Equal)
Angle CBE = angle D（
be equal to
）
Δ ABP ∽ DCP (three internal angles are equal)
AP*CP=BP*DP（
）
EB * EA = EC * ed（
）
EF*EF= EB*EA=EC*ED（
）
（
,
,
(the circular curtain theorem)
AB*CD+AD*CB=AC*BD（
Ptolemy）

On the sine theorem, In the triangle ABC, if a / cosa = B / CoSb, then the shape of the triangle is________ . (isosceles triangle, I can understand why it turns to the end, Tana = tanb, and why can't a + B be equal to 180?)