Four-point * * * Circle There are several basic methods to prove a four-point * * circle: Method 1 Choose three points from the four points of a circle to make a circle, and then prove that the other 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 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 (the circumferential angles of the same arc are equal), we can confirm the four points of the circle. (If we can prove that their vertex angles are right angles, we can confirm the four points of the circle, and the connecting line between the two points on the hypotenuse is the diameter of the circle. ) Method 3 connects 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 4 connects the four points of the proved * * * circle into two intersecting line segments. If it can be proved that the products obtained by dividing the two line segments by their intersection points are equal, the four points of the * * * circle can be affirmed (according to the inverse theorem of the intersection theorem); 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 be sure that these four points are also * * * circles. (According to the inverse theorem of Ptolemy's theorem) Method 5 proves that the distances between the points of the proved * * * circle and a certain point are all equal, thus judging that they are * * * circles. Because the vertical lines of the three sides of the quadrilateral connected together have intersections, these four-point circles can be confirmed. The foundation of each of the above five basic methods is one of the reasons for the four-point circle, so when it is necessary to prove the four-point circle, choose one of these five basic methods to prove it. Judgement 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, the intersection of AB and DC is extended to E, the intersection E is the tangent EF of the circle O, and AC and BD intersect at P, then A+C=π, B+D=π, and the angle DBC= angle d AC (the circumferential angles of the same arc are equal). Angle CBE= Angle ADE (external angle equals internal angle) △ABP∽△DCP (three internal angles are equal) AP*CP=BP*DP (chord theorem) EB*EA=EC*ED (secant theorem) EF*EF= EB*EA=EC*ED (secant theorem).