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The orthocentroidal circle of a triangle DeltaABC is a central circle having the segment joining the triangle centroid G and orthocenter H of DeltaABC as its diameter ...

For a right triangle with legs a and b and hypotenuse c, a^2+b^2=c^2. (1) Many different proofs exist for this most fundamental of all geometric theorems. The theorem can ...

The symmedial triangle DeltaK_AK_BK_C (a term coined here for the first time), is the triangle whose vertices are the intersection points of the symmedians with the reference ...

The lines AK_A, BK_B, and CK_C which are isogonal to the triangle medians AM_A, BM_B, and CM_C of a triangle are called the triangle's symmedian. The symmedians are ...

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, ...

The orthic axis of the excentral triangle, which is central line L_1 (Casey 1888, p. 177; Kimberling 1998, p. 150) and therefore has trilinear equation alpha+beta+gamma=0. It ...

Two lines PQ and RS are said to be antiparallel with respect to the sides of an angle A if they make the same angle in the opposite senses with the bisector of that angle. If ...

A polygon which has both a circumcircle (which touches each vertex) and an incircle (which is tangent to each side). All triangles are bicentric with R^2-x^2=2Rr, (1) where R ...

A bicentric quadrilateral, also called a cyclic-inscriptable quadrilateral, is a four-sided bicentric polygon. The inradius r, circumradius R, and offset x are connected by ...

The dual of Pascal's theorem (Casey 1888, p. 146). It states that, given a hexagon circumscribed on a conic section, the lines joining opposite polygon vertices (polygon ...