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)
|
Similarly, for the line segment 
in space determined by 
and 
, the midpoint can be calculated as
 |
(2)
|
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)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
(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
![Delta_P=Delta_Q=Delta[1-((m_1)/(a_1)+(m_2)/(a_2)+(m_3)/(a_3))+(m_2m_2)/(a_2a_3)+(m_3m_1)/(a_3a_1)+(m_1m_2)/(a_1a_2)],](/images/equations/Midpoint/NumberedEquation3.svg) |
(6)
|
where 
are the sides of the original triangle
and 
are the lengths of the triangle
medians (Johnson 1929).
