This write-up is based on an exploration of any triangle (RST) and its medial triangle (DEF). A medial triangle is one that is formed by connecting the midpoints of the sides of the original triangle. It is similar to the original triangle and one-fourth of its area.
Specifically, the exploration focuses on how the centroid (G), orthocenter (H), circumcenter (C), and incenter (I) of the original triangle and the medial triangle compares for different shaped triangles.
For each case shown below, G, H, C, and I were used to represent the special points for the large triangle RST. Points G', H'. I', and C' were used for the medial triangle DEF.
As I began the exploration and constructed the centroid (G), I noticed this point was concurrent fro both triangles and remained that way with different shaped triangles. Therefore, only one label (G) was used for both triangles. this can be seen in all the cases shown below.
The construction of the orthocenter of the medial triangle shows the point concurrent with that of the circumcenter for triangle RST and remains when exploring different shaped triangles. Notice that C and H' are concurrent for all the cases shown below.
click here to investigate GSP sketch
centroid (G) concurrent
orthocenter (H') of triangle DEF is concurrent with circumcenter (C) of triangle RST
incenters (I and I') are not concurrent
centers of the triangles remain interior of their respective triangles.
centroid (G) is concurrent
orthocenter (H') and circumcenter (C) remain concurrent
orthocenter falls on vertex angle of right triangle
circumcenter falls at midpoint of the hypotenuse
all points are collinear for this case
all points are concurrent in the equilateral triangle
orthocenter and circumcenter points are exterior their respective triangles
all points are collinear in this case
orthocenter and circumcenter are exterior of their respective triangles
points are not collinear
Summary
The centroid remains concurrent for all cases.
The circumcenter of triangle RST and the orthocenter of triangle DEF remain concurrent for all cases.
All the points are collinear given any isosceles triangle.
All points are concurrent given any equilateral triangle.
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