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The midpoint theorem generalizes to the intercept theorem, where rather than using midpoints, both sides are partitioned in the same ratio. [1] [2] The converse of the theorem is true as well. That is if a line is drawn through the midpoint of triangle side parallel to another triangle side then the line will bisect the third side of the triangle.
The interior perpendicular bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the circumcenter (the center of the circle through the three vertices). Thus any line through a ...
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.
Consider a triangle ABC.Let the angle bisector of angle ∠ A intersect side BC at a point D between B and C.The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of the length of side AB to the length of side AC:
The triangle medians and the centroid.. In geometry, a median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. . Every triangle has exactly three medians, one from each vertex, and they all intersect at the triangle's cent
Any median (which is necessarily a bisector of the triangle's area) is concurrent with two other area bisectors each of which is parallel to a side. [1] A cleaver of a triangle is a line segment that bisects the perimeter of the triangle and has one endpoint at the midpoint
Each cleaver through the midpoint of one of the sides of a triangle is parallel to the angle bisectors at the opposite vertex of the triangle. [1] [2] The broken chord theorem of Archimedes provides another construction of the cleaver. Suppose the triangle to be bisected is ABC, and that one endpoint of the cleaver is the midpoint of side AB.
In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. [1] [2] Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name. [3]