For three points on the plane that are not on the same straight line, there must be a circle to make these three points on the circumference, so "three-point * * * circle" is meaningless.

And "four-point * * * circle" means that for four points, there is a circle that makes all four points on the circumference. Any four points do not meet this condition.

"Four-point * * * circle" has three properties: (1) The vertex angles of two triangles connected by four points of * * * circle are equal; (2) Diagonal complementation of quadrilateral inscribed in a circle; (3) The outer angle of a quadrilateral inscribed with a circle is equal to the inner diagonal.

What is a four-point circle?

On the basis of the nature of four-point * * * circle, the judgment of four-point * * * circle is proved.

The properties of four-point * * * circle;

(1) The circumferential angles of the same arc are equal.

(2) Diagonal complementation of quadrilateral inscribed in a circle

(3) The outer angle of a quadrilateral inscribed with a circle is equal to the inner diagonal.

According to the fact that the angle of the circle is equal to half of the arc it subtends, the above properties can be proved.

Judgement theorem of four-point * * * circle;

Method 1 Connect the four points of the proved * * * circle into two triangles with * * * as the base, and both triangles are on the same side of the base. If we can prove that the vertex angles are equal, we can be sure of the four points of the circle.

(It can be said that if the angle between two points on the same side of the line segment and two points at both ends of the line segment is equal, then these two points and four points at both ends of the line segment are * * * circles. )

Method 2: Connect the four points of the proved circle into a quadrilateral. If it can be proved that its diagonal is redundant or that one of its outer angles is equal to the inner diagonal of its adjacent complementary angle, then the four points of the * * * circle can be sure.

It can be said that if four points on a plane are connected into a quadrilateral, then the complementary angle or an outer angle of the diagonal is equal to its inner diagonal. Then these four dots * * * circle)

We can all use a method in mathematics; Prove by reducing to absurdity.

Now, if four points on the plane are connected into a quadrilateral, they are diagonally complementary. Then these four dots * * * circle "are proved as follows (draw a proof diagram for others as follows).

It is known that in quadrilateral ABCD, ∠ A+∠ C = 180.

Prove: quadrilateral ABCD inscribed circle (A, B, C, D four-point * * * circle).

Prove: by reducing to absurdity

Make a circle o through a, b and d, assuming that c is not on the circle o, but c is outside or inside the circle,

If C is outside the circle, let BC and O intersect at C' and connect DC', and according to the properties of the quadrilateral inscribed in the circle, we can get ∠ A+∠ DC' B = 180.

∠∠a+∠c = 180 ∴∠dc'b=∠c

This contradicts the triangle exterior angle theorem, so C can't be outside the circle. Similarly, it can be proved that C can't be in a circle.

∴C is on the circle O, that is, on the four-point * * * circle of A, B, C and D.

What do the four * * * circles of A, B, C and D mean?

A, b, c and d are on the same circle.

For example, taking the diagonal intersection o of rectangular ABCD as the center,

Make a circle o with OA as the radius, OA = OB = OC = OD,

∴A, B, C and D are all on ⊙ O,

It's called a, b, c and d four-point circle.

The third grade mathematics is shown in the picture. What does APDH mean by a four-point circle? Connect four points to form a circle? Overseas emergency action

But these four points can be on the same circle.