Find the angle of triangle OMB1.

[NagatoKami]

On the sides BC and AC of triangle ABC, equilateral triangles BCA1 and ACB1 are constructed externally. Point M is the midpoint of side AB and O is the center of triangle BCA1. Find the angles of triangle OMB1.

 

[Dr.Peterson]

On the sides BC and AC of triangle ABC, equilateral triangles BCA1 and ACB1 are constructed externally. Point M is the midpoint of side AB and O is the center of triangle BCA1. Find the angles of triangle OMB1.

I would start by drawing a better picture, that is at least somewhat to scale. (Your equilateral triangles don't look equilateral, and that may affect your thinking.)

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[khansaheb]

......and O is the center of triangle BCA1....

There are 4 "most popular centers" of a triangle- centroid, incenter, circumcenter, and orthocenter.

Which "one" are you referring to here?

 

[pka]

There are 4 "most popular centers" of a triangle- centroid, incenter, circumcenter, and orthocenter.

Which "one" are you referring to here?

In an equilateral triangle those are one and the same point. SEE HERE

 

[khansaheb]

In an equilateral triangle those are the one and the same point. SEE HERE

I know that - I was trying to get OP to start thinking about the propertie/s of center/s.

 

[Dr.Peterson]

I am waiting to hear something about the context of the problem, and what sort of thinking is available (e.g. geometry of similar triangles, trigonometry, etc.), but here is one step that I find helpful for several possible approaches: Add in some more midpoints and associated line segments:

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I do see some similar triangles, which are provable by labeling various sides and angles. And I do have at least one full solution. But how you do it is up to you.