Pedal Triangles

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This assignment will explore the various pedal triangles of a given triangle. And, compare the smiliarities of a subordinate pedal triangle. Let's first recall how to construct a pedal triangle -- given any triangle and any point in the plane of the triangle, there exists three lines through the pedal point, P, that intersect each side of the given triangle at a perpendicular angle.

We will investigate the pedal triangle of the pedal triangle of the pedal triangle of the pedal point P of triangle ABC. Observe the construction of this subordinate pedal triangle:

The pink triangle, A'B'C', is similar to our given triangle ABC. How?

Well, ALL the corresponding angles are congruent and the sides are proportional to each other. Observe the following GSP calculations:

To see how this similiarity holds when the pedal point P is manipulated (translated) in GSP, click here. (GSP 4.06 software required)

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Now, let's consider a new given triangle XYZ and a new pedal point, D, in the plane:

We can manipulate the size of triangle XYZ (i.e. acute, obtuse, right) and the location of the pedal point D (i.e. interior, exterior, on the sides, at the vertices) in order to discover the change of the pedal triangle (the yellow triangle). To manipulate, click here. (GSP 4.06 software required)

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Remark: When the pedal point D is either the centroid, incenter, orthocenter, or circumcenter, the pedal triangle has the following interesting observations:

Centroid -- pedal triangle location is limitless except where the centroid corresponds to the vertices, at these points the pedal triangle is undefined

Incenter -- pedal triangle is always on the interior of the given triangle regardless of its size and at the vertices it is undefined

Orthocenter -- pedal triangle is limitless except where the orthocenter corresponds to the vertices, at these points the pedal triangle is undefined

Circumcenter -- pedal triangle location is limitless (the circumcenter can never correspond to the vertices, so the pedal triangle is always defined)

To witness these observations first hand, please click my class page below and use the GSP Script Tools link to facilitate the construction of a triangle's centroid, incenter, orthocenter, and circumcenter, respectively.

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