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Two triply perspective triangles BbY and CcX. If two triangles are a central couple in at least two different ways (with two different associations of corresponding vertices, and two different centers of perspectivity) then they are perspective in three ways. This is one of the equivalent forms of Pappus's (hexagon) theorem. [5]
By definition, two triangles are perspective if and only if they are in perspective centrally (or, equivalently according to this theorem, in perspective axially). Note that perspective triangles need not be similar .
Two triangles and are said to be in perspective centrally if the lines , , and meet in a common point, called the center of perspectivity. They are in perspective axially if the intersection points of the corresponding triangle sides, X = A B ∩ a b {\displaystyle X=AB\cap ab} , Y = A C ∩ a c {\displaystyle Y=AC\cap ac} , and Z = B C ∩ b c ...
For a given triangle ABC, let H A H B H C be its orthic triangle and T A T B T C the triangle formed by the outer tangents to its three excircles.These two triangles are similar and the Clawson point is their center of similarity, therefore the three lines T A H A, T B H B, T C H C connecting their vertices meet in a common point, which is the Clawson point.
The triangle ABC and the cevian triangle A'B'C' are in perspective and let DEF be the axis of perspectivity of the two triangles. The line DEF is the trilinear polar of the point Y. The line DEF is the central line associated with the triangle center X.
This composition is a bijective map of the points of S 2 onto itself which preserves collinear points and is called a perspective collineation (central collineation in more modern terminology). [7] Let φ be a perspective collineation of S 2. Each point of the line of intersection of S 2 and T 2 will be fixed by φ and this line is called the ...
Specifying two sides and an adjacent angle (SSA), however, can yield two distinct possible triangles unless the angle specified is a right angle. Triangles are congruent if they have all three sides equal (SSS), two sides and the angle between them equal (SAS), or two angles and a side equal (ASA) (Book I, propositions 4, 8, and 26).
Let ABC be any triangle. Let the Euler line of triangle ABC meet the sidelines BC, CA and AB of triangle ABC at D, E and F respectively. Let A g B g C g be the triangle formed by the Euler lines of the triangles AEF, BFD and CDE, the vertex A g being the intersection of the Euler lines of the triangles BFD and CDE, and similarly for the other two vertices.