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In any triangle, if one of the sides is extended, the exterior angle is greater than both the interior and opposite angles.

The first Morley center is the center of Morley's circle. It has triangle center function alpha_(356)=cos(1/3A)+2cos(1/3B)cos(1/3C) and is Kimberling center X_(356).

The half-altitude circle is the circumcircle of the half-altitude triangle. The circle function is given by l=(f(a,b,c))/(128a^2b^3c^3cosAcosBcosC), (1) where (2) which is ...

The isogonal transform of a geometric object is the object obtained by collectively taking the isogonal conjugates of all its points.

If P is any point on a line TT^' whose orthopole is S, then the circle power of S with respect to the pedal circle of P is a constant (Gallatly 1913, p. 51).

The MacBeath circle, a term coined here for the first time, is the circumcircle of the MacBeath triangle. It has a fairly complicated radius, center function, and circle ...

Morley's circle is the circumcircle of the first Morley triangle. Its center is the first Morley center, which has center function alpha_(356)=cos(1/3A)+2cos(1/3B)cos(1/3C), ...

Given an acute angle in a right triangle, the opposite side is the leg of the triangle which is not incident on the angle. Lengths of opposite and adjacent sides appear ...

Consider a reference triangle DeltaABC and any given point P. The perpendiculars to AP, BP and CP respectively meet BC, AC and AB in three collinear points defining line l. ...

In the plane, if a line intersects one side of a triangle and misses the three vertices, then it must intersect one of the other two sides. This is a special case of the ...