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The triangle's nine-point circle has half the diameter of the circumcircle. In any given triangle, the circumcenter is always collinear with the centroid and orthocenter. The line that passes through all of them is known as the Euler line. The isogonal conjugate of the circumcenter is the orthocenter.
Triangles constructed on the unit circle can also be used to illustrate the periodicity of the trigonometric functions. First, construct a radius OP from the origin O to a point P( x 1 , y 1 ) on the unit circle such that an angle t with 0 < t < β π / 2 β is formed with the positive arm of the x -axis.
The three splitters of a triangle all intersect each other at the Nagel point of the triangle. A cleaver of a triangle is a segment from the midpoint of a side of a triangle to the opposite side such that the perimeter is divided into two equal lengths. The three cleavers of a triangle all intersect each other at the triangle's Spieker center.
In geometry, the circumference (from Latin circumferens, meaning "carrying around") is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment . [ 1 ]
As proved by Archimedes, in his Measurement of a Circle, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, [11] which comes to π multiplied by the radius squared: =.
The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's medial triangle (with vertices at the midpoints of the sides of the reference triangle) and its orthic triangle (with vertices at the feet of the reference triangle's altitudes). [6]: p.153
Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of dπ at the centre of the circle), each with an area of β 1 / 2 β · r 2 · dπ (derived from the expression for the area of a triangle: β 1 / 2 β · a · b · sinπ ...
The formulas and properties given below are valid in the convex case. The word cyclic is from the Ancient Greek κΟκλος (kuklos), which means "circle" or "wheel". All triangles have a circumcircle, but not all quadrilaterals do. An example of a quadrilateral that cannot be cyclic is a non-square rhombus.