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