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In any triangle, the locus of a point whose pedal triangle has a constant Brocard angle and is described in a given direction is a circle of the Schoute coaxal system.

The Neuberg A_1-circle is the locus of the polygon vertex A_1 of a triangle on a given base A_2A_3 and with a given Brocard angle omega. From the center N_1, the base A_2A_3 ...

The Kiepert hyperbola is a hyperbola and triangle conic that is related to the solution of Lemoine's problem and its generalization to isosceles triangles constructed on the ...

A number which is simultaneously a nonagonal number N_m and a square number S_n and therefore satisfies the Diophantine equation 1/2m(7m-5)=n^2. (1) Completing the square and ...

The radical circle of the Lucas circles is the circumcircle of the Lucas tangents triangle. Its center has trilinear center function alpha_(1151)=2cosA+sinA (1) corresponding ...

The Gallatly circle is the circle with center at the Brocard midpoint X_(39) and radius R_G = Rsinomega (1) = (abc)/(2sqrt(a^2b^2+a^2c^2+b^2c^2)), (2) where R is the ...

Given the incircle and circumcircle of a bicentric polygon of n sides, the centroid of the tangent points on the incircle is a fixed point W, known as the Weill point, ...

An addition-multiplication square is a square of integers that is simultaneously a magic square and multiplication magic square. In 1955, Horner found a square of order eight ...

The line on which the orthocenter H, triangle centroid G, circumcenter O, de Longchamps point L, nine-point center N, and a number of other important triangle centers lie. ...

A right pyramid is a pyramid for which the apex lies directly above the centroid of the base. A regular pyramid is therefore a special case of a right pyramid.