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The nine-point circle is tangent to the incircle and excircles. In 1822 Karl Feuerbach discovered that any triangle's nine-point circle is externally tangent to that triangle's three excircles and internally tangent to its incircle; this result is known as Feuerbach's theorem. He proved that:
Thus, the nine-point center forms the center of a point reflection that maps the medial triangle to the Euler triangle, and vice versa. [citation needed] According to Lester's theorem, the nine-point center lies on a common circle with three other points: the two Fermat points and the circumcenter. [9] The Kosnita point of a triangle, a ...
The nine-point circle is tangent to the incircle and excircles. In geometry, the nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are: [28] [29] The midpoint of each side of the triangle; The foot ...
In geometry, the Euler line, named after Leonhard Euler (/ ˈ ɔɪ l ər / OY-lər), is a line determined from any triangle that is not equilateral.It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter, the circumcenter, the centroid, the Exeter point and the center of the nine-point circle of the triangle.
In particular, if the points are diametrically opposite, their Simson lines are perpendicular and in this case the intersection of the lines lies on the nine-point circle. Letting H denote the orthocenter of the triangle ABC, the Simson line of P bisects the segment PH in a point that lies on the nine-point circle.
Common nine-point circle, where N, O 4, A 4 are the nine-point center, circumcenter, and orthocenter respectively of the triangle formed from the other three orthocentric points A 1, A 2, A 3. The center of this common nine-point circle lies at the centroid of the four orthocentric points. The radius of the common nine-point circle is the ...
Nine-point circle (centered at nine-point center N which, along with H, G, O, lies on the Euler line e) In geometry , a triangle center or triangle centre is a point in the triangle 's plane that is in some sense in the middle of the triangle.
The orthocenter H, the centroid G, the circumcenter O, and the center N of the nine-point circle all lie on a single line, known as the Euler line. [18] The center of the nine-point circle lies at the midpoint of the Euler line, between the orthocenter and the circumcenter, and the distance between the centroid and the circumcenter is half of ...