Square, bisector, and circle surprise#mathematics #math #maths#geometry #stem

Added: 2026-05-17

[00:00:00]Check this out. This is pretty crazy.

[00:00:02]Start with a square and draw ray that bisects an angle there. See how we have an angle bisector there? This point is movable anywhere. We can move it anywhere just as long as we have an angle bisector up there. See what I mean? Two equal angles. Next, where that angle where that angle bisector hits the right side of the square, we're going to drop a perpendicular to that other ray right there. And next, we're going to draw a circle that passes through the square's four vertices.

[00:00:30]It turns out if you do this, something interesting occurs.

[00:00:34]The lower left vertex of the square, that orange point, and that uh point in which the perpendicular hits, and the point at which that segment hits the circle, believe it or not, those three points are always collinear. They will always line up, believe it or not.

[00:00:52]Pretty neat, right? So, it doesn't matter where that angle bisector is, and of course, it doesn't matter how big or how small the square is anyway.

[00:01:00]But, here we will always have three collinear points. The three orange points will always lie on a line, or they'll always line up, or the slope between the first two and the slope between the second two and the slope between the first and the third, they will all be the same. [music] But, why does this happen? How can we formally prove that true?

[00:01:20]Any ideas? Any thoughts?

[00:01:23]Feel free to drop them right in the comments.