Consider the acute triangle ABC below along with it's circumcircle, points of

intersection with altitudes, and orthocenter H.

There are numerous relationships to notice. To point out and persuade about a few

details lets also examine the triangle constructed from where the extended altitudes

meet the circumcircle and create triangle RQP.

If we construct another triangle with points EFD we notice that the points are simply

the midpoints of segments of equal length that are intersections of triangles ABC and

RQP.

This will be the orthic triangle below

Will this always hold?

Will pointsEFD always be midpoints of their respective segmenst?

Click here to make some general observations when working with the obtuse situation.

(Hint: grab any angle and pull it out greater than 90 degrees)

We see that if the triangle is right triangle PQR and EFD become the degenerate

triangle.

and if ABC is obtuse, the relationship of the orthic triangle and the midpoints of the

intersecting triangle segments fail to exist and we simply have EDF outside of the

scenario and having different relations altogether.

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