Circle • The midpoint of any
diameter of a
circle is the center of the circle. • Any line
perpendicular to any
chord of a circle and passing through its midpoint also passes through the circle's center. • The
butterfly theorem states that, if is the midpoint of a
chord of a
circle, through which two other chords and are drawn; and intersect chord at and correspondingly, then is the midpoint of .
Ellipse • The midpoint of any segment which is an
area bisector or
perimeter bisector of an
ellipse is the ellipse's center. • The ellipse's center is also the midpoint of a segment connecting the two
foci of the ellipse.
Hyperbola • The midpoint of a segment connecting a
hyperbola's vertices is the center of the hyperbola.
Triangle • The
perpendicular bisector of a side of a
triangle is the line that is perpendicular to that side and passes through its midpoint. The three perpendicular bisectors of a triangle's three sides intersect at the
circumcenter (the center of the circle through the three vertices). • The
median of a triangle's side passes through both the side's midpoint and the triangle's opposite
vertex. The three medians of a triangle intersect at the triangle's
centroid (the point on which the triangle would balance if it were made of a thin sheet of uniform-density metal). • The
nine-point center of a triangle lies at the midpoint between the circumcenter and the
orthocenter. These points are all on the
Euler line. • A
midsegment (or
midline) of a triangle is a line segment that joins the midpoints of two sides of the triangle. It is parallel to the third side and has a length equal to one half of that third side. • The
medial triangle of a given triangle has vertices at the midpoints of the given triangle's sides, therefore its sides are the three midsegments of the given triangle. It shares the same centroid and medians with the given triangle. The
perimeter of the medial triangle equals the
semiperimeter (half the perimeter) of the original triangle, and its area is one quarter of the area of the original triangle. The
orthocenter (intersection of the
altitudes) of the medial triangle coincides with the
circumcenter (center of the circle through the vertices) of the original triangle. • Every triangle has an
inscribed ellipse, called its
Steiner inellipse, that is internally tangent to the triangle at the midpoints of all its sides. This ellipse is centered at the triangle's centroid, and it has the largest area of any ellipse inscribed in the triangle. • In a
right triangle, the circumcenter is the midpoint of the
hypotenuse. • In an
isosceles triangle, the median,
altitude, and perpendicular bisector from the
base side and the
angle bisector of the
apex coincide with the Euler line and the
axis of symmetry, and these coinciding lines go through the midpoint of the base side.
Quadrilateral • The two
bimedians of a
convex quadrilateral are the line segments that connect the midpoints of opposite sides, hence each bisecting two sides. The two bimedians and the line segment joining the midpoints of the diagonals are
concurrent at (all intersect at)a point called the "vertex centroid", which is the midpoint of all three of these segments. • The four "maltitudes" of a convex quadrilateral are the perpendiculars to a side through the midpoint of the opposite side, hence bisecting the latter side. If the quadrilateral is
cyclic (inscribed in a circle), these maltitudes all meet at a common point called the "anticenter". •
Brahmagupta's theorem states that if a cyclic quadrilateral is
orthodiagonal (that is, has
perpendicular diagonals), then the perpendicular to a side from the point of intersection of the diagonals always goes through the midpoint of the opposite side. •
Varignon's theorem states that the midpoints of the sides of an arbitrary quadrilateral form the vertices of a
parallelogram, and if the quadrilateral is not self-intersecting then the area of the parallelogram is half the area of the quadrilateral. • The
Newton line is the line that connects the midpoints of the two diagonals in a convex quadrilateral that is not a parallelogram. The line segments connecting the midpoints of opposite sides of a convex quadrilateral intersect in a point that lies on the Newton line.
General polygons • A
regular polygon has an
inscribed circle which is
tangent to each side of the polygon at its midpoint. • In a regular polygon with an even number of sides, the midpoint of a
diagonal between opposite vertices is the polygon's center. • The
midpoint-stretching polygon of a
cyclic polygon (a
polygon whose vertices all fall on the same circle) is another cyclic polygon inscribed in the same circle, the polygon whose vertices are the midpoints of the
circular arcs between the vertices of . Iterating the midpoint-stretching operation on an arbitrary initial polygon results in a sequence of polygons whose shapes converge to that of a
regular polygon. ==Generalizations==