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,
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.
to investigate GSP sketch
orthocenter (H') of triangle
DEF is concurrent with circumcenter (C)
of triangle RST
and I') are not concurrent
centers of the triangles remain
interior of their respective triangles.
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
all points are concurrent in the
orthocenter and circumcenter points
are exterior their respective triangles
all points are collinear in this
orthocenter and circumcenter are
exterior of their respective triangles
points are not collinear
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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