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Ordinary trigonometry studies triangles in the Euclidean plane .There are a number of ways of defining the ordinary Euclidean geometric trigonometric functions on real numbers, for example right-angled triangle definitions, unit circle definitions, series definitions [broken anchor], definitions via differential equations [broken anchor], and definitions using functional equations.
The Simson line of a vertex of the triangle is the altitude of the triangle dropped from that vertex, and the Simson line of the point diametrically opposite to the vertex is the side of the triangle opposite to that vertex. If P and Q are points on the circumcircle, then the angle between the Simson lines of P and Q is half the angle of the ...
Then the triangle formed by the lines A H A O, B H B O, C H C O and triangle ABC are homothetic and congruent, and the homothetic center lies on the line OH. Dao Thanh Oai's result is generalization of all results above. [6] [7] [8] When HO is the Euler line, Dao's result is the Gossard perspector theorem.
Fig. 1 – A triangle. The angles α (or A), β (or B), and γ (or C) are respectively opposite the sides a, b, and c.. In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles.
Second generalization: Let a conic S and a point P on the plane. Construct three lines d a , d b , d c through P such that they meet the conic at A, A'; B, B' ; C, C' respectively. Let D be a point on the polar of point P with respect to (S) or D lies on the conic (S).
In Euclidean geometry, Kosnita's theorem is a property of certain circles associated with an arbitrary triangle. Let A B C {\displaystyle ABC} be an arbitrary triangle, O {\displaystyle O} its circumcenter and O a , O b , O c {\displaystyle O_{a},O_{b},O_{c}} are the circumcenters of three triangles O B C {\displaystyle OBC} , O C A ...
In 2000, Bernard Gibert proposed a generalization of the Lester Theorem involving the Kiepert hyperbola of a triangle. His result can be stated as follows: Every circle with a diameter that is a chord of the Kiepert hyperbola and perpendicular to the triangle's Euler line passes through the Fermat points.
Generalization for arbitrary triangles, green area = blue area Construction for proof of parallelogram generalization. Pappus's area theorem is a further generalization, that applies to triangles that are not right triangles, using parallelograms on the three sides in place of squares (squares are a special case, of course). The upper figure ...