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 we can prove this, we can determine these four points.

Method 2

If it can be proved that the four points of a circle are right angles, it can be determined that these four points are * * * circles.

Method 3

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.

Method 4

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.

Method 5

Connect the four points of the proved * * * circle into two intersecting line segments. If we can prove that the products of these two line segments divided by the intersection points are equal, we can determine that these four points are * * * circles; 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.

Method 6

The distance between points of a * * * circle is equal to a certain point, which determines their * * * circles.

Just choose method four.

∠O 1PR=∠O 1RP，∠O2QS=∠O2SQ，from ∠O3RS=∠O3SR。

Get ∠O 1RS=∠O2SR。

There are ∠ o1pr+∠ o2sr+∠ o2sq = ∠ o1RP+∠ o1RS+∠ o2qs.

Now we have to prove that these three angles add up to 180 degrees, or ∠O 1PR+∠O2QS=∠O3SR=∠O3RS.

And ∠O3O 1O2=2∠O 1PR, ∠ O3O 1 = 2 ∠ O2QS, so ∠ o2o3o1=/kloc-0.

Therefore ∠ O3 Sr = (180-∠ o2o 3o1)/2 = ∠ o1pr+∠ o2qs.

Therefore, diagonal equality and complementarity are obtained.