Concerning the planimetry problem: Having a suitable background, it is not hard to produce many puzzles like this. For example, here is one more pair of points on that mysterious circle: intersect a circle centered at $B$ with radius $BM$ and a circle centered at $A$ with radius $AB$, then the two intersection points are also on that 8-points circle.

This circle is a circle of Apollonius with foci at $A$ and $B$. More precisely, it is the locus of points $X$ such that $BX:AX=\sin\alpha$ where $\alpha=\angle BAM=\frac12\angle BAC$. The fact that this is a circle orthogonal to the original one is a general property of circles of Apollonius, and verifying that the ratio of distances equals $\sin\alpha$ for each point is straightforward.

Concerning the planimetry problem: Having a suitable background, it is not hard to produce many puzzles like this. For example, here is one more pair of points on that mysterious circle: intersect a circle centered at $B$ with radius $BM$ and a circle centered at $A$ with radius $AB$, then the two intersection points are also on that 8-points circle.

This circle is a circle of Apollonius with foci at $A$ and $B$. More precisely, it is the locus of points $X$ such that $BX:AX=\sin\alpha$ where $\alpha=\angle BAM=\frac12\angle BAC$. The fact that this is a circle orthogonal to the original one is a general property of circles of Apollonius, and verifying that the ratio of distances equals $\sin\alpha$ for each point is straightforward.