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Concurrent lines arise in the dual of Pappus's hexagon theorem. For each side of a cyclic hexagon, extend the adjacent sides to their intersection, forming a triangle exterior to the given side. Then the segments connecting the circumcenters of opposite triangles are concurrent. [8]
Ceva's theorem, case 1: the three lines are concurrent at a point O inside ABC Ceva's theorem, case 2: the three lines are concurrent at a point O outside ABC. In Euclidean geometry, Ceva's theorem is a theorem about triangles.
Given any line l, let P, Q, R be the feet of perpendiculars from the vertices A, B, C of triangle ABC to l. The lines through P. Q, R perpendicular respectively to the sides BC, CA, AB are concurrent and the point of concurrence is the orthopole of the line l with respect to the triangle ABC. In modern triangle geometry, there is a large body ...
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]
The vertices of every triangle fall on a circle called the circumcircle. (Because of this, some authors define "concyclic" only in the context of four or more points on a circle.) [2] Several other sets of points defined from a triangle are also concyclic, with different circles; see Nine-point circle [3] and Lester's theorem.
The line joining them is then called the Pascal line of the hexagon. Brianchon: If all six sides of a hexagon are tangent to a conic, then its diagonals (i.e. the lines joining opposite vertices) are three concurrent lines. Their point of intersection is then called the Brianchon point of the hexagon.
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:
From any point on the circumcircle of a triangle, the nearest points on each of the three extended sides of the triangle are collinear in the Simson line of the point on the circumcircle. The lines connecting the feet of the altitudes intersect the opposite sides at collinear points. [3]: p.199