Definition: ABC is a triangle; A′B′C′ is its medial triangle; L, M, N are the feet of the perpendiculars from A, B, C respectively on a straight line, σ1 σ2, σ3 are three circles having their centres at A′, B′, C′ respectively such that σ1 passes through M and N, σ2 through N and L, and σ3 through L and M Then it is readily seen that the radical axes of the three circles taken in pairs meet in a point W—the radical centre of the circles. Again, since L is a common point of σ2 and σ3, their radical axis is the line through L at right angles to their line of centres B′C′, and hence to BC. Hence the lines through L, M, N at right angles to BC, CA, AB respectively meet at W, which is said to be the orthopole of the line LMN The common radical circle of σ1, σ2, σ3 is called the Orthopolar circle of the line LMN. Let us consider a few applications of the properties of this circle and its centre.