Midpoint
The point on a Line Segment dividing it into two segments of equal length. The midpoint of a line segment iseasy 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). Itis more challenging to locate the midpoint using only a Compass, but Pedoe (1995, pp. xviii-xix) gives one solution.
In a Right Triangle, the midpoint of the Hypotenuse is equidistant from the three Vertices (Dunham 1990).
Given a Triangle 
with Area 
, locate the midpoints 
. Nowinscribe two triangles 
and 
with Vertices 
and 
placed so that 
. Then and have equal areas
where
are the sides of the original triangle and 
are the lengths of theMedians (Johnson 1929).See also Archimedes' Midpoint Theorem, Brocard Midpoint, Circle-Point Midpoint Theorem, Line Segment, Median (Triangle), Midpoint Ellipse
References
Dunham, W. Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 120-121, 1990. Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, p. 80, 1929. Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.
- Cusp Form
- Cusp Map
- Cusp Point
- Cutting
- Cut-Vertex
- CW-Approximation Theorem
- CW-Complex
- Transfinite Diameter
- Transfinite Number
- Transform
- Transformation
- Transitive
- Transitive Closure
- Transitive Reduction
- Transitivity Class
- Translation
- Translation Relation
- Transpose
- Transpose Map
- Transposition
- Transposition Group
- Transposition Order
- Transversal Array
- Transversal Design
- Transversal Line