Explanation for this method to find incenter of triangle

Question

Euclidea is a mobile game that requires you to construct certain geometric structures using only a straightedge and a compass.

One of the levels requires you to consruct the incenter of a scalene triangle within some stipulated moves. While scouring the internet, I came across this solution(the second method, starting from 0:27).

I have spent a good amount of time trying to understand the solution to no avail (especially the rescaling of the triangle) so I would be very grateful if anyone could explain the solution in a clear, step-wise manner.

Answer 1

The solution shown in the video is doing nothing more than constructing the perpendicular bisector of a side of an isosceles triangle whose apex angle is the same as the angle we want to bisect.

Refer to the figure above. When the circle with center $A$ and radius $AC$ is drawn, it intersects $AB$ at $D$, so that $AC = AD$ and $\triangle ACD$ is isosceles. Then $\angle CAB = \angle CAD$, and shares the same angle bisector $AF$. But bisecting this angle is the same as constructing the perpendicular bisector of side $CD$. How would you do that? You'd just draw two circles with equal radii at centers $C$ and $D$ whose radius is more than half the distance between them. These circles intersect at $F$.

Note that we do not need to choose $AC$ as the radius of circle $A$, nor do we need to choose $AC = AD$ as the radius of circles $C$ and $D$. These choices were made simply because these points already existed in the diagram.

Answer 2

Hint

$AE$ is the radical axis of the red and green circles ($\implies AE \perp BD$). Furthermore, $\triangle ABD$ is isosceles. These are enough to show that $AE$ is an angle bisector. Can you take it from here?

I'll be happy to answer your questions, but please show more effort (so that the question doesn't remain as a link-only problem) in the original post.