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: Altitudes

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Michael de Villiers, created 1 September 2021.