Consider a point F and the lines joining it with the vertices of triangle ABC. Consider also a point D moving on line FC, and the three medial lines of the triangle. Draw the orthogonal to FC at D cutting two of the medial lines at G and E respectively. Draw then the perpendiculars from E, G to the lines FA and FB. Show that these lines intersect at a point I on the third medial line of the triangle.
Hint: Draw the circles with centers at E, G and radii EC, GC respectively. They intersect lines FA, FB at second points A', B' respectively. The circle through BB'A'A is centered at I. That I moves on a line e, through the circumcenter follows from the similarity of all triangles EGI. I being equidistant from A, B shows that e is the third medial line.
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Pedal triangle problem ![[0_0]](PedalProblem_files/PedalProblem_0_0_0.png)
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