The Orthopolar Circle

Extract

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.