.: Orthocenter of a Triangle (remarkable point) :.
Orthocenter of a Triangle (remarkable point):
Dados quaisquer três pontos não colineares A, B e C, o baricentro do triângulo ABC
é o ponto G que é o encontro (único) das alturas do triângulo.
Uma altura é a reta que passa por um dos vértices (A) e cruza o lado oposto do triângulo fazendo ângulo reto (BC).
Como existem 3 cevianas do tipo altura, podem ser tomadas as interseções entre elas 2-a-2 (pA com pB; pA com pC; e pB com pC), resultando 3 possível interseções.
Entretanto estas 3 interseções sempre coincidem! Notável, não?
remarkable point
An remarkable Point of a triangle is the point resulting of the intersection of three lines
that passes through each triangle vertex and its opposite side (cevian lines).
The remarkable fact is the existence of a single point, since there are 3 possible intersections
(one to each pair of lines)!
These lines is named as cevians in honor the mathematician Giovanni Ceva (that published a condition to existence of this single point in 1678).
See: H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967.
Fig.: static figure representing the Gergonne (note that I is the incenter) point (an special "remarkable point").
Interactive construction with iGeom
Below is presented the orthocenter point of a (generic) Triangle in iGeom.
Hands on: try to move the point A (B or C) and observe that:
the altitude lines has a common point O as its intersection.
In order to move any point (the best) option is: with the "move button"
selected (a single right "click" on it), make a right "click" over
the target point (this means "click" and release the mouse button), then
freely move the mouse around the drawing area.
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