Let perpendiculars AD, BE, and CF be drawn to a straight line m from the vertices of triangle ABC. From point D, the foot of the perpendicular from A, draw line DX perpendicular to side BC, the side of triangle ABC opposite angle A. Similarly, draw line EY perpendicular to side AC and FZ perpendicular to side AB. Then, the perpendiculars from D, E, and F concur at a point called the orthopole of line m and triangle ABC.
Click here for a GSP sketch of the orthopole.
How would you prove the perpendiculars from points D, E, and F are concurrent? Give it a try on your own. If you need help click here for a proof.
An Interesting Property of the Orthopole
The orthopole of a line through the circumcenter lies on the nine-point circle of the triangle.
Try using GSP and construct the situation describe by this property. Use the GSP utility on my web page to constuct a nine-point circle given a triangle. This will make it much easier to construct.
How would you prove this? Give it a try.
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