Altitudes and Orthocenters

LetŐs
first construct a triangle ABC, with orthocenter H.

Now,
letŐs construct three more triangles from the orthocenter to the vertices of
the larger triangle.

Now,
letŐs construct the orthocenters of triangle ACH, triangle ABH and triangle
BCH.

Notice
that the orthocenter of triangle ACH is point B, the orthocenter of triangle
ABH is point C, and the orthocenter of triangle BCH is point A.

Now,
letŐs construct the circumcircle of all four triangles: triangle ABC, triangle
ACH, triangle ABH and triangle BCH.

It
appears as if the four circumcircles are congruent, letŐs find out.

Yes,
using GSP measurement tools, I know that the circumcircles are congruent. Now
letŐs see what will happen if I created a triangle using the centers of the
colored circumcircles as vertices.

Wow,
this triangle is congruent to our original triangle ABC.

Conjectures:

Given
a triangle ABC and its orthocenter H, we can construct three triangles,
triangle ACH, triangle ABH and triangle BCH such that:

1.
The
circumcircles of all four triangles are congruent.

2.
The
intersection of any two of the circumcircles belonging to the smaller triangles
is a vertex of the given triangle ABC.

3.
The
orthocenters of the smaller triangles is a vertex of the larger triangle.

4.
A
triangle formed by connecting the centers of the circumcircles of the smaller
triangles is congruent to the original triangle ABC.