All articles

Ernst Cassirer: Neo-Kantianism and mathematics | Tangente

Cassini I, founder of the Paris Observatory | Tangente

Cassini IV and the map of France | Tangente

Jacques Cassini and Earth's shape | Tangente

César-François Cassini: A nationwide map of France | Tangente

Cassini family: French astronomers | Tangente

Star puzzle: assembling polyhedra | Tangente

Felice Casorati and essential singularities | Tangente

Mary Cartwright: dynamical systems and chaos | Tangente

Hélène Cartan: first woman admitted to ENS Ulm | Tangente

Control chart: statistical quality tool | Tangente

Henri Cartan: Bourbaki and complex functions | Tangente

Adams circle

Consider a triangle ABC and its incircle, tangent to side BC at D, to side AB at E and to side AC at F. The lines AD, BF and CE are concurrent at a particular point of the triangle, called the Gergonne point. Draw through this point the lines parallel to the three sides of the triangle. These parallels intersect the sides of the triangle at six points, denoted I, J, K, L, M and N, which prove to be concyclic. The circle through these six points is called the Adams circle, after the Swiss mathematician Karl Adams (1811–1849), who proved that these six points are concyclic.

circle of Apollonius

The term circle of Apollonius denotes two distinct notions. First, the locus of points M such that the ratio MA/MB is constant and equal to k (with k > 0 and k ≠ 1) is a circle, called the circle of Apollonius associated with points A and B and the ratio k. This circle has the segment IJ as its diameter, where I and J are the points on line AB that divide the segment BA in the ratio k, internally and externally. Second, the fractal figure generated by three mutually tangent circles is also called the circle of Apollonius, or sometimes the Apollonian gasket. Starting with a curvilinear triangle whose sides are arcs of circles, we inscribe a circle tangent to these three arcs, then insert new circles into the gaps by repeating the process indefinitely. This construction is one of the oldest known examples of a fractal, studied by Apollonius of Perga in the third century BCE.

radical center

The radical center of three circles is the point with the same power with respect to each of the three circles. It is the intersection of the three pairwise radical axes: the radical axis of two circles is the locus of points with the same power with respect to each of them. If the three circles have non-collinear centers, their radical center exists and is unique. The radical center is a fundamental tool in the study of pencils of circles and circle configurations in classical geometry.

Incenter

In triangle geometry, the incenter is the intersection point of the triangle's three internal angle bisectors. This point, called the incenter and often denoted I, is equidistant from the triangle's three sides. It is the centre of the unique circle tangent to the three sides and lying inside the triangle, called the incircle. The incenter is always located inside the triangle, regardless of the type of triangle.

Excenter

The excenter opposite A in triangle ABC is the intersection of the internal bisector of angle A and the external bisectors of angles B and C. This point is the center of a circle tangent to the triangle's three sides but lying outside it. Every triangle has exactly three excircles, one for each vertex. The distance from vertex A to the point where the excircle touches side AB (or side AC) equals the triangle's semiperimeter.

circumcenter

The circumcenter of a triangle is the point of intersection of the three perpendicular bisectors of its sides. This point, often denoted O, is equidistant from the triangle's three vertices: it is the center of the unique circle passing through these three vertices, called the circumcircle of the triangle. If the triangle is right-angled, this center is the midpoint of the hypotenuse. If the triangle is acute, it lies inside the triangle; if it is obtuse, it lies outside.

center of the nine-point circle

The nine-point circle is a well-known circle associated with every triangle, passing through nine particular points: the midpoints of the three sides, the feet of the three altitudes, and the midpoints of the segments joining the orthocenter H to each of the three vertices. This circle is the image of the circumcircle under the homothety centered at G (the centroid) with ratio -1/2, and also under the homothety centered at H (the orthocenter) with ratio 1/2. Its center, denoted E, therefore lies on the Euler line, at the midpoint of the segment OH. Moreover, GE = (1/3) GH.

center of symmetry

A center of symmetry of a figure is a point O such that the figure is invariant under a half-turn about O. In other words, for every point M of the figure, its image M' under reflection in O also belongs to the figure. A figure having a center of symmetry is said to be symmetric about that point. When a figure has several axes of symmetry, their point of intersection is a center of symmetry of the figure. For example, the center of a circle is a center of symmetry; the center of a parallelogram is the intersection of its diagonals; and the center of a rectangle or a rhombus is the intersection of its axes of symmetry.