# Calculate Third Point of Triangle knowing Vector

I have 3 points and their coordinates $$(A, B, C)$$. Then I have new coordinates of points $$A'$$ and $$B'$$ How to calculate the coordinates of point $$C'$$ knowing that the distance from point A to point C and the angle between line AB and line AC is the same. It must work as a formula in any program or on a sheet of paper.

Available

Point $$A$$ (the $$x$$ and $$y$$ coordinate) Point $$B$$ (the $$x$$ and $$y$$ coordinate) Point $$C$$ (the $$x$$ and $$y$$ coordinate) Point $$A'$$ (the $$x$$ and $$y$$ coordinate) Point $$B'$$ (the $$x$$ and $$y$$ coordinate)

$$|A−C|=|A′−C′|$$ but $$|A−B|$$ don't equal $$|A′−B′|$$

Angle between $$AC$$ and $$AB$$ is the same as $$A'C'$$ and $$A'B'$$

Unknown

All Angles

Point $$C'$$ (the $$x$$ and $$y$$ coordinate)

My conclusions

$$AB=\sqrt{(Bx−Ax)^2+(By−Ay)^2}$$

$$AC=\sqrt{(Cx−Ax)^2+(Cy−Ay)^2}$$

$$CB=\sqrt{(Bx−Cx)^2+(By−Cy)^2}$$

$$A'B'=\sqrt{(B'x−A'x)^2+(B'y−A'y)^2}$$

• In the triangle formed by A', B' & C' we only know one side and one angle. So the given information is insufficient to find the coordinates of C'. – Wishwas Jul 4 at 8:34
• But we also know the distance of Point C''; from Point A'';, which gives sufficient information to create a vector at the end of which Point C''; will be located. I just don't know how to make it. – Joachim Biernacki Jul 4 at 8:43
• Step 1. Shift origin to A'. Step2. Rotate Axis so that A'B' is along X-axis. Now A'C' is a line making a known angle with X-axis. Now C' a point on this line at a known distance from the current origin. So you get its co-ordinates. Now reverse the rotation in step 2. Reverse the shift of origin done in step 1. – Wishwas Jul 4 at 9:28
• I don't understand how to do it in maths. – Joachim Biernacki Jul 4 at 9:46