In the plane Cartesian coordinate system, it is judged whether the four known points form a square

If the coordinates of 4 points are known, assume that point A is (x1, y1), b(x2, y2), c(x3, y3), d(x4, y4). Yes.

Let the coordinates of the three points A, B, and C be known in the plane Cartesian coordinate system, and if you want to find the coordinates of the D point of the square ABCD, 1. First, check that the known points should meet the requirements that ABC is an isosceles right triangle, otherwise there is no solution. Assuming that b is a right-angled vertex, 2. write the equations for the straight line ab and bc respectively;

3. According to the equal slope of the two parallel lines, write the straight line equation L1 parallel to BC through point A and the straight line equation L2 parallel to AB through point C in point oblique formula;

4. Find the coordinates of the intersection of the straight line L1 and L2, which is the coordinates of point D.

1. Ibid.

2. Find the coordinates of the midpoint O of the AC line and the length of AO, 3. Write the circular equation with O as the center and AO as the radius; Write the equation for the straight line bo;

4. Find the intersection point of the circle and the straight line Bo, and take the one different from the point B, and get the coordinates of point D.

Calculate the length of AB, BC, AC respectively to see if it conforms to the Pythagorean theorem, or whether there are two equal Take the midpoint D of BC, connect AD, use the midpoint coordinate formula to calculate the midpoint coordinates, and then use the distance formula to bring in the coordinates of a and D to calculate the length of AD.

Right-angled triangle.

bc = 4 + 36 = 40 = 2 10ac = 1 + 49 = 5 2 under the root

ab = 1 + 9 = 10 under the root number

Because AB side + BC side = AC side.

According to the Pythagorean theorem, the triangle ABC is a right triangle.

According to the Pythagorean theorem:

oa=ob=5,oc=od=10,a, b, c, d are not on the same circle centered on o.

First of all, draw a coordinate axis, take a(-2 6,1) as an example, as ae, af are perpendicular to the x-axis, and the y-axis (ae x-axis, af y-axis) is connected with ao, then there is |ae|=2√6，|af|=1, then |ao|=√【（2√6）^2+1^2】=√25=5

ae|"It's an absolute value" ^ "is the meaning of square) In the same way, it can be calculated that b, c, d 3 points, and the distance from the origin o are all 5, so that a, b, c, d 4 points are all on the circle with o as the center and 5 as the radius.

AB two points are symmetrical with respect to the bisector of the two quadrant angles, and CD two points are symmetrical with respect to the x-axis, so to make the four points co-circular, the center of the circle must be the coordinate origin (the intersection of the perpendicular lines of the two points), and OA=ob=5, OC=OD=10, so the four points are not conspecific.

If oa=ob=oc=od, then the four points a, b, c, and d are on the same circle with o as the center, and if they are not equal, a, b, c, and d are not on the same circle with o as the center.