Knowra Orthocenter Orthocenter The orthocenter is the point where a triangle’s three altitudes meet. Its position depends on the triangle’s angles: it can lie inside, on, or outside the triangle.
Altitude (triangle) : A perpendicular segment or line from a triangle’s vertex to the line containing the opposite side. Each altitude is one of the three lines whose common point defines the orthocenter.
Perpendicular lines : Lines that intersect to form a right angle. An altitude is defined by perpendicularity to the opposite side’s line.
Euler line : The line through a triangle’s circumcenter, centroid, and orthocenter, when these points are distinct. The orthocenter lies on this line with two other major triangle centers.
Equilateral triangle : A triangle with three equal sides and three equal 60-degree angles. Its orthocenter coincides with its centroid, circumcenter, and incenter.
Incenter : The common intersection of a triangle’s three internal angle bisectors and the center of its incircle. Unlike the orthocenter, it is defined by angle bisectors and always lies inside the triangle.
Ceva's theorem : A theorem giving a condition for three cevians of a triangle to be concurrent. It provides a general concurrency framework for the triangle’s three altitudes.
Triangle : A polygon with three sides, three vertices, and three interior angles. The orthocenter is defined from the three altitudes of this basic figure.
Centroid : The common intersection of a triangle’s three medians. Its alignment with the orthocenter and circumcenter forms the Euler line.
Isosceles triangle : A triangle with at least two equal sides. Its symmetry axis contains the orthocenter.
Excenter : A center of a triangle’s excircle, formed by intersections of angle bisectors, including external bisectors. Its construction contrasts with altitude concurrency and can place a triangle center outside the triangle.
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