Question

how to inscribe a semi circle in an equilateral triangle

Answer 1

Since equilateral triangle has all sides equal, it is symmetric.  SO it is easy to find the incenter.

Draw perpendicular bisectors or the sides.  Or, draw perpendicular from each vertex on to the opposite sides.  Or, join midpoint of each side with the opposite vertex.

The point of intersection will be the incenter as well as circum center also.

Now with the radius as the length between incenter and the mid-points of sides, draw a circle.

A semicircle only touches two sides.  So start at the midpoint of base and draw up to the point when it meets the altitude from the top vertex.