Power Lines Special Case: Altitudes of a Triangle

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

Related Links
Power Lines of a Triangle
Water Supply II: Three Towns (Rethinking Proof activity)
The Center of Gravity of a Triangle (Rethinking Proof activity)
Triangle Altitudes (Rethinking Proof activity)
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Napoleon (Rethinking Proof activity)
Miquel (Rethinking Proof activity)
Kosnita's Theorem
Fermat-Torricelli Point Generalization (aka Jacobi's theorem) plus Further Generalizations
Dual to Kosnita (De Villiers points of a triangle)
Another concurrency related to the Fermat point of a triangle plus related results
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva's theorem)
Point Mass Centroid (centre of gravity or balancing point) of Quadrilateral
Bride's Chair Concurrency
Triangle Centroids of a Hexagon form a Parallelo-Hexagon: A generalization of Varignon's Theorem
Nine-point centre & Maltitudes of Cyclic Quadrilateral
A side trisection triangle concurrency
A 1999 British Mathematics Olympiad Problem and its dual
Concurrency and Euler line locus result
Haag Hexagon - Extra Properties
Concurrency, collinearity and other properties of a particular hexagon
Carnot's Perpendicularity Theorem & Some Generalizations
Generalizing the concepts of perpendicular bisectors, angle bisectors, medians and altitudes of a triangle to 3D
Anghel's Hexagon Concurrency theorem
Some Circle Concurrency Theorems
Three Overlapping Circles (Haruki's Theorem)
Parallel-Hexagon Concurrency Theorem
Toshio Seimiya Theorem: A Hexagon Concurrency result
Van Aubel's Theorem and some Generalizations (See concurrency in Similar Rectangles on sides)
The quasi-circumcentre and quasi-incentre of a quadrilateral

External Links
Concurrent lines (Wikipedia)
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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