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  2. Cyclic quadrilateral - Wikipedia

    en.wikipedia.org/wiki/Cyclic_quadrilateral

    The direct theorem was Proposition 22 in Book 3 of Euclid's Elements. [3] Equivalently, a convex quadrilateral is cyclic if and only if each exterior angle is equal to the opposite interior angle . In 1836 Duncan Gregory generalized this result as follows: Given any convex cyclic 2 n -gon, then the two sums of alternate interior angles are each ...

  3. Circumcircle - Wikipedia

    en.wikipedia.org/wiki/Circumcircle

    The circumcenter is the point of intersection between the three perpendicular bisectors of the triangle's sides, and is a triangle center. More generally, an n -sided polygon with all its vertices on the same circle, also called the circumscribed circle, is called a cyclic polygon , or in the special case n = 4 , a cyclic quadrilateral .

  4. Triangle center - Wikipedia

    en.wikipedia.org/wiki/Triangle_center

    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. For example, the centroid, circumcenter, incenter and orthocenter were familiar to the ancient Greeks, and can be obtained by simple constructions.

  5. Nine-point center - Wikipedia

    en.wikipedia.org/wiki/Nine-point_center

    A triangle showing its circumcircle and circumcenter (black), altitudes and orthocenter (red), and nine-point circle and nine-point center (blue) In geometry, the nine-point center is a triangle center, a point defined from a given triangle in a way that does not depend on the placement or scale of the triangle.

  6. Euler line - Wikipedia

    en.wikipedia.org/wiki/Euler_line

    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.

  7. Difference of two squares - Wikipedia

    en.wikipedia.org/wiki/Difference_of_two_squares

    Galileo's law of odd numbers. A ramification of the difference of consecutive squares, Galileo's law of odd numbers states that the distance covered by an object falling without resistance in uniform gravity in successive equal time intervals is linearly proportional to the odd numbers. That is, if a body falling from rest covers a certain ...

  8. de Longchamps point - Wikipedia

    en.wikipedia.org/wiki/De_Longchamps_point

    As the reflection of the orthocenter around the circumcenter, the de Longchamps point belongs to the line through both of these points, which is the Euler line of the given triangle. Thus, it is collinear with all the other triangle centers on the Euler line, which along with the orthocenter and circumcenter include the centroid and the center ...

  9. Encyclopedia of Triangle Centers - Wikipedia

    en.wikipedia.org/wiki/Encyclopedia_of_Triangle...

    Each point in the list is identified by an index number of the form X(n) —for example, X(1) is the incenter. [6] The information recorded about each point includes its trilinear and barycentric coordinates and its relation to lines joining other identified points. Links to The Geometer's Sketchpad diagrams are provided for key points. The ...

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