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If the radii are equal, the radical axis is the line segment bisector of M 1, M 2. In any case the radical axis is a line perpendicular to ¯. On notations. The notation radical axis was used by the French mathematician M. Chasles as axe radical. [1] J.V. Poncelet used chorde ideale. [2]
Let be the circumcircle of triangle and be the tangency point of the -mixtilinear incircle and . Let X ≠ T A {\displaystyle X\neq T_{A}} be the intersection of line T A D {\displaystyle T_{A}D} with Γ {\displaystyle \Gamma } and Y ≠ T A {\displaystyle Y\neq T_{A}} be the intersection of line T A E {\displaystyle T_{A}E} with Γ ...
So the orthic axis of ABC is the trilinear polar of the orthocenter of ABC. The orthic axis of ABC is the axis of perspectivity of ABC and its orthic triangle H A H B H C. It is also the radical axis of the triangle's circumcircle and nine-point-circle.
Figure 1: The point O is an external homothetic center for the two triangles. The size of each figure is proportional to its distance from the homothetic center. In geometry, a homothetic center (also called a center of similarity or a center of similitude) is a point from which at least two geometrically similar figures can be seen as a dilation or contraction of one another.
Construction of the Malfatti circles: [3] For a given triangle determine three circles, which touch each other and two sides of the triangle each. Spherical version of Malfatti's problem: [4] The triangle is a spherical one. Essential tools for investigations on circles are the radical axis of two circles and the radical center of three circles.
The radical axis of a pair of circles is defined as the set of points that have equal power h with respect to both circles. For example, for every point P on the radical axis of circles 1 and 2, the powers to each circle are equal: h 1 = h 2. Similarly, for every point on the radical axis of circles 2 and 3, the powers must be equal, h 2 = h 3.
The line connecting these common intersection points is the radical axis for all three circles. The two isodynamic points are inverses of each other relative to the circumcircle of the triangle. The centers of these three circles fall on a single line (the Lemoine line ).
The radical axis of two circles is the set of points of equal tangents, or more generally, equal power. Circles may be inverted into lines and circles into circles. [clarification needed] If two circles are internally tangent, they remain so if their radii are increased or decreased by the same amount.