Post by blair on Jan 25, 2020 20:30:43 GMT
I think the following is a simpler solution to the problem of a rectangle inscribed in a circle.
Points A, B, C, D are on the unit circle in the complex plane, such that A + B + C + D = 0. Rotate the circle so that line AB is parallel to the real axis. Due to reflection symmetry about the imaginary axis, line AB is centered on the imaginary axis, with the real parts of points A and B adding to zero.
It follows that the real parts of line CD must also add to zero, therefore that line is also parallel to the real axis and centered on the imaginary axis. So far, it may have a different length.
If we repeat this procedure for line BC, we will see that line DA is also parallel to it. Thus we now have a parallelogram, so side AB is the same length as the opposite side CD. As sides AB and CD are both centered on the imaginary axis, the only possible shape that can connect them is a rectangle.
I hope you like this.
Blair Dowden
Points A, B, C, D are on the unit circle in the complex plane, such that A + B + C + D = 0. Rotate the circle so that line AB is parallel to the real axis. Due to reflection symmetry about the imaginary axis, line AB is centered on the imaginary axis, with the real parts of points A and B adding to zero.
It follows that the real parts of line CD must also add to zero, therefore that line is also parallel to the real axis and centered on the imaginary axis. So far, it may have a different length.
If we repeat this procedure for line BC, we will see that line DA is also parallel to it. Thus we now have a parallelogram, so side AB is the same length as the opposite side CD. As sides AB and CD are both centered on the imaginary axis, the only possible shape that can connect them is a rectangle.
I hope you like this.
Blair Dowden
