Search Results for ""

A special case of Apollonius' problem requiring the determination of a circle touching three mutually tangent circles (also called the kissing circles problem). There are two ...

The conjecture that, for any triangle, 8omega^3<ABC (1) where A, B, and C are the vertex angles of the triangle and omega is the Brocard angle. The Abi-Khuzam inequality ...

In the figure above with tangent line PT and secant line PA, (PA)/(PT)=(PT)/(PB) (1) (Jurgensen et al. 1963, p. 346). The line tangent to a circle of radius a centered at ...

Given a triangle DeltaA_1A_2A_3, the points A_1, I, and J_1 lie on a line, where I is the incenter and J_1 is the excenter corresponding to A_1. Furthermore, the circle with ...

The inner Napoleon circle, a term coined here for the first time, is the circumcircle of the inner Napoleon triangle. It has center at the triangle centroid G (and is thus ...

The Longuet-Higgins circle is the radical circle of the circles centered at the vertices A, B, and C of a reference triangle with respective radii b+c, c+a, and a+b. Its ...

The outer Napoleon circle, a term coined here for the first time, is the circumcircle of the outer Napoleon triangle. It has center at the triangle centroid G (and is thus ...

Divide a triangle by its three medians into six smaller triangles. Surprisingly, the circumcenters O_(AB), O_(BA), etc. of the six circumcircles of these smaller triangles ...

The probability P(a,n) that n random arcs of angular size a cover the circumference of a circle completely (for a circle with unit circumference) is ...

The circumcircle, Brocard circle, Lemoine axis, and isodynamic points belong to a coaxal system orthogonal to the Apollonius circles, called the Schoute coaxal system. In ...