In geometry, the orthopole of a system consisting of a triangle ABC and a line ℓ in the same plane is a point determined as follows. [1] Let A ′, B ′, C ′ be the feet of perpendiculars dropped on ℓ from A, B, C respectively. Let A ′′, B ′′, C ′′ be the feet of perpendiculars dropped from A ′, B ′, C ′ to the sides opposite A, B, C (respectively) or to those sides' extensions. Then the three lines A ′ A ′′, B ′ B ′′, C ′ C ′′, are concurrent. [2] The point at which they concur is the orthopole.
Due to their many properties, [3] orthopoles have been the subject of a large literature. [4] Some key topics are determination of the lines having a given orthopole [5] and orthopolar circles. [6]
In geometry, the orthopole of a system consisting of a triangle ABC and a line ℓ in the same plane is a point determined as follows. [1] Let A ′, B ′, C ′ be the feet of perpendiculars dropped on ℓ from A, B, C respectively. Let A ′′, B ′′, C ′′ be the feet of perpendiculars dropped from A ′, B ′, C ′ to the sides opposite A, B, C (respectively) or to those sides' extensions. Then the three lines A ′ A ′′, B ′ B ′′, C ′ C ′′, are concurrent. [2] The point at which they concur is the orthopole.
Due to their many properties, [3] orthopoles have been the subject of a large literature. [4] Some key topics are determination of the lines having a given orthopole [5] and orthopolar circles. [6]