There are several basic methods to prove a four-point * * * circle: Method 1 First, select three points from the four points of the proved * * * circle to make a circle, and then prove that another point is also on this circle. If this can be proved, the four-point circle is certain. Method 2 Connect the four points of the proved * * * circle into two triangles at the bottom of * * *.

(If it can be proved that the two vertex angles are right angles, it can be determined that these four points are * * * circles, and the connecting line between the two points on the hypotenuse is the diameter of the circle. ) method 3. Connect the four points of the proved * * * circle into a quadrilateral. If it can be proved that the diagonals are complementary or that one of its outer angles is equal to the inner diagonal of its adjacent complementary angles, the four-point * * * circle can be affirmed. Method four. Connect the four points of the proved * * * circle into two intersecting line segments. If it can be proved that the products of these two line segments divided by their intersection points are equal, the four-point * * circle can be affirmed. Or connect the four points of the proved * * * circle in pairs and extend the two intersecting line segments. If we can prove that the product of two line segments from the intersection to the two endpoints of one line segment is equal to the product of two line segments from the intersection to the two endpoints of another line segment, we can confirm that these four points are also * * * circles. (According to the inverse theorem of Ptolemy's theorem) Method 5 proves that the distances from the points of the proved * * * circles to a certain point are all equal, thus determining them * *.

The foundation of each of the above five basic methods is one of the reasons for the four-point circle. Therefore, when it is required to prove the four-point circle problem, we should first choose one of the five basic methods to prove it according to the conditions of the proposition and the characteristics of the graph.

Judgment and nature:

The sum of diagonals of a quadrilateral inscribed in a circle is 180, and any external angle is equal to its internal angle.

If the quadrilateral ABCD is inscribed in the circle O, extending the intersection of AB and DC to E, the intersection E is the tangent EF of the circle O, and AC and BD intersect at P, then A+C=π, and B+D=π.

Angle DBC= Angle DAC (the circumferential angle of the same arc is equal).

Angle CBE= angle ADE (outer angle equals inner diagonal)

△ABP∽△DCP (three internal angles are equal)

AP*CP=BP*DP (Chord Theorem)

Picture of a four-point * * * circle EB*EA=EC*ED (Secant Theorem)

EF*EF=

EB*EA=EC*ED (cutting line theorem)

(Secant Theorem, Secant Theorem, and Intersection Theorem are collectively referred to as circular power theorem)

AB*CD+AD*CB=AC*BD (Ptolemy theorem)

Alternating line segment theorem

Method 6

Theorem of four-vertex * * * circles of two RT triangles with the same hypotenuse, whose hypotenuse is the diameter of the circle * * * Theorem of four-point * * * circle Method of judging four-point * * * circle 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 their 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

If the four points of the proved * * * circle are connected into a quadrilateral, if it can be proved that the diagonals are complementary or that one of its outer angles is equal to the inner diagonal of its neighbor's complementary angle, then the four-point * * * circle can be affirmed.

It can be said that if four points on a plane are diagonally complementary or an outer angle is equal to its inner diagonal. Then the four-point reduction to absurdity proves that it is now "if four points on a plane are diagonally complementary in a quadrilateral." Then this four-point * * * circle is proved as follows (others draw a proof diagram 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 and point c is outside or inside the circle,

If point C is outside the circle, let BC and circle O intersect at C' and connect DC'. According to the properties of quadrilateral inscribed in a 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.