How to find the coordinates of the vertex of a triangle if the coordinates of the other two and the lengths of all sides are known?

Question

Mathematics

There is a triangle ABC. The coordinates of points A and B are known, as well as the lengths of all sides. The sides AC and BC are equal.
How to find the coordinates of point C? Can anyone help?

Answer 1

There are many options. For example:
1. Find the midpoint of segment AB, let's call point S.
2. Find the height of the triangle. Given that the triangle is isosceles, the height is the perpendicular bisector.
3. Construct a vector parallel to AB with length equal to the height of the triangle.
4. Rotate it 90 degrees to the left or right.
5. Move the beginning of the vector to point S.

Answer 2

for example

Answer 3

1) We build a circle with radius AC from point A 2) We build a circle with radius BC from point B 3) We solve the system of equations, we get 0 (no intersections), 1 (intersection at one point, touching) or 2 real roots (intersection at 2x points). These are the possible options for point C. You can find x , and then substitute in any of the equations and get y , or vice versa.
(x-xA)^2 + (y-yA)^2 = R^2 = AC^2
(x-xB)^2 + (y-yB)^2 = R^2 = BC^2