Editing Midpoint (section)

===Triangle===

*The [[bisection#Triangle|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 (geometry)|median]] of a triangle's side passes through both the side's midpoint and the triangle's opposite [[vertex (geometry)|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 [[altitude]]s) of the medial triangle coincides with the [[circumcenter]] (center of the circle through the vertices) of the original triangle.

*Every triangle has an [[inscribed figure|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 (triangle)|altitude]], and perpendicular bisector from the [[base (geometry)|base]] side and the [[angle bisector]] of the [[Apex (geometry)|apex]] coincide with the Euler line and the [[axis of symmetry]], and these coinciding lines go through the midpoint of the base side.