PEDAL TRIANGLES and the Nine-Point Circle

Alone at nights,  I read my Bible more and Euclid less.

Robert Buchanan (1841-1901)
(An Old Dominie's Story)

Let triangle ABC be any triangle. If P is any point in the plane, then the triangle formed by constructing perpendiculars to the sides of ABC (extended if necessary) from P, where points R, S, and T are the intersections, is called the Pedal Triangle.

Triangle RST is the Pedal Triangle for Pedal Point P.

Click here for a GSP script to construct a pedal triangle where P is any point in the plane of ABC.

Click here for a GSP animation for various locations of point P.


What if ... pedal point P is the center of the nine point circle for triangle ABC?

Recall the Nine-Point Circle for any triangle is the circle that contains the midpoints of the three sides, the midpoints of the lines joining the orthocenter to the three vertices, and the feet of the three altitudes. The following is an illustration of the Nine-Point Circle.


Now, construct the pedal triangle RST using the center, P, of the nine-point circle for triangle ABC as the pedal point.

 

Will the pedal triangle RST ever lie on the Nine-Point Circle?

Yes! Notice the Pedal Triangle RST will lie on the nine-point circle when the circumcenter, CC, orthocenter, H, and the pedal point, P, all are the same point; that is, when the circumcenter and the orthocenter become the center of the nine point circle, also known as P.

Here are a few examples:








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