## The Symmedian or Lemoine Centre

The reflection of a line through a vertex in the bisector of the angle at that vertex is called its **isogonal** line.
Thus AX and AY are isogonals if BAX = CAY or BAY = CAX. The isogonals of three concurrent lines are themselves concurrent.
These pairs of points are called **isogonal conjugates** of each other.

The isogonals of the medians, i.e. their reflections in the angle bisectors, are called the **symmedians** and they
meet at the **Lemoine centre**, G' (the isogonal conjugate of the centroid G).

The symmedians divide the sides in the ratio of the squares of the other sides, i.e. BL' = a.c²/(c² + b²),
CL' = a.b²/(c² + b²) etc BL'/CL' = c²/b².

The distances of G' from the sides are in the ratio of the sides, a:b:c.
The Lemoine point G' is the point at which the areal coordinates are such that p² + q² + r² is minimum.
Thus G' is the centre of mass of masses a², b², c², at A, B, C.
The areas BG'C, CG'A, AG'B are in the ratios a²:b²:c².

The triangle P'Q'R' where the symmedians meet the circumcircle has the same symmedians as ABC.

If squares are drawn outwards on the sides of a triangle and the outer sides of the squares are extended to form a
triangle the centre of homothecy of the two triangles is the Lemoine centre of each of them. Conversely we can start with the
outer triangle and construct the inner.