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, σ 1 σ 2 , σ 3 are three circles having their centres at A ′, B ′, C ′ respectively such that σ 1 passes through M and N , σ 2 through N and L , and σ 3 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 σ 2 and σ 3 , 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 σ 1 , σ 2 , σ 3 is called the Orthopolar circle of the line LMN . Let us consider a few applications of the properties of this circle and its centre.