trisecting an arbitrary angle

Question

I've read and seen in a video that it's suppose to be impossible to trisect an arbitrary angle with just a compass and straight-edge. This seems fairly simple though. I'm not sure I understand this correctly. Can someone nicely explain why this doesn't work? Am I misunderstanding the problem? Trisecting a circle

Answer 1

It isn't easy to tell from the video, but towards the end when the display shows that the polygons are equal I take this to mean that the areas of the polygons (i.e. the three triangles) are equal.

If the above is true, then what you have done is trisect a line segment that is a chord of a circle centred on the vertex of the angle. To trisect an arbitrary angle you instead need to trisect an arc of a circle centred on the vertex of the angle.

To illustrate I've chosen a (non-arbitrary) right angle.

The dotted lines pass through points that trisect the chord.

The solid lines pass through points that trisect the arc, and the illustrated angles are between the axes and these solid lines.

It is easily seen that the two constructions are not the same.

It is a well-known and mathematically proven result that an arbitrary angle cannot be trisected using just a compass and straightedge.