Consider triangle ABC with respective midpoints D, E and F of its sides as shown below. Now note that the Power Lines theorem applied to triangle DEF immediately implies the concurrency of the altitudes of triangle ABC (since the corresponding sides of EFD are parallel to those of ABC). Alternatively, from a higher viewpoint, one could say that in this case the point of concurrency, the power point or radical centre, of the three radical axes of the three circles, is coincident with the orthocentre of triangle ABC.
Power Lines Special Case: AltitudesChallenge: Can you prove the concurrency of the altitudes with regard to the construction given in the above sketch without using the Power Lines theorem (or Carnot's Perpendicularity theorem)?
Michael de Villiers, created 1 September 2021.