Search results
Results From The WOW.Com Content Network
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]
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct lines , which either is one point (sometimes called a vertex ) or does not exist (if the lines are parallel ).
That is, in triangles ABC and DEF with sides a, b, c, and d, e, f respectively (with a opposite A etc.), if a = d and b = e and angle C > angle F, then >. The converse also holds: if c > f, then C > F. The angles in any two triangles ABC and DEF are related in terms of the cotangent function according to [6]
As shown in the accompanying animation, the theorem can be proved using similar triangles. In the version illustrated here, the triangle A B C {\displaystyle \triangle ABC} gets reflected across a line that is perpendicular to the angle bisector A D {\displaystyle AD} , resulting in the triangle A B 2 C 2 {\displaystyle \triangle AB_{2}C_{2 ...
This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude. [23] The length of the altitude is the distance between the base and the vertex. The three altitudes intersect in a single point, called the orthocenter of the triangle. [24]
In an axiomatic formulation of Euclidean geometry, such as that of Hilbert (modern mathematicians added to Euclid's original axioms to fill perceived logical gaps), [1]: 108 a line is stated to have certain properties that relate it to other lines and points. For example, for any two distinct points, there is a unique line containing them, and ...
The lines A 1 A 2, B 1 B 2 drawn through corresponding endpoints of those radii, which are homologous points, intersect each other and the line of centers at the external homothetic center. Conversely, the lines A 1 B 2 , B 1 A 2 drawn through one endpoint and the opposite endpoint of its counterpart intersects each other and the line of ...
(Let D be the vertex opposite to the side formed by the tangent at the vertex A, E be the vertex opposite to the side formed by the tangent at the vertex B, and F be the vertex opposite to the side formed by the tangent at the vertex C.) The lines through DA', EB', FC' are concurrent. The point of concurrence is the Exeter point of ABC.