check out the many references on constructing a perpendicular bisector of a segment. Your problem does not specify a perpendicular bisector, but that is the easiest way to find the midpoint of a line segment.
Draw your angle and a line traversing its sides. Then just construct the perpendicular bisector of that segment.
For a point, give a construction of a line so the portion intercepted by the segments of a given angle are bisected at the point.
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Thank you.
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