The point on a line segment dividing it into two segments of equal length. The midpoint of a line segment is easy to locate by first constructing a lens using circular arcs, then connecting the cusps of the lens. The point where the cusp-connecting line intersects the segment is then the midpoint (Pedoe 1995, p. xii). It is more challenging to locate the midpoint using only a compass (i.e., a Mascheroni construction).
For the line segment in the plane determined by and , the midpoint can be calculated as
(1)
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Similarly, for the line segment in space determined by and , the midpoint can be calculated as
(2)
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In a right triangle, the midpoint of the hypotenuse is equidistant from the three polygon vertices (Dunham 1990).
In the figure above, the trilinear coordinates of the midpoints of the triangle sides are , , and .
The midpoint of a line segment with endpoints and given in trilinear coordinates is , where
(3)
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(4)
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(5)
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(left as an exercise in Kimberling 1998, p. 35, Ex. 15).
Given a triangle with area , locate the midpoints of the sides . Now inscribe two triangles and with polygon vertices and placed so that . Then and have equal areas
(6)
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where are the sides of the original triangle and are the lengths of the triangle medians (Johnson 1929).