Two perpendicular segments ME and NF are drawn to the tangent EF of the circle centered at point O from the ends of the diameter MN. prove that the tangency point P divides the segment EF in half.
Since the segments ME and NF are perpendicular to the tangent EF, the quadrilateral MEFH is a rectangular trapezoid.
The line segment OP is the radius of the circle drawn to the tangency point, then OP is perpendicular to the tangent EF.
Since ME, OP and HF are perpendicular to EF, they are parallel.
Point O is the center of the circle, that is, the middle of the diameter MH, then OP is the middle line of the trapezoid, and therefore point P is the middle of EF, which was required to prove.
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