Assignment9

I construct the pedal triangle below.
The yellow triangle (RST) is the pedal
triangle.
You can see and move this triangle
by clicking here.
In this file, I made a script tool
for construction of a pedal triangle (please look at the bottom
of the tool box in that file and find the word "Pedal Triangle"
in the middle of the box). You choose any three points that are
supposed to be verticles of the first trianle, and then you choose
on the same plane another point that is supposed to be the point
P. The set of these four points leads the pedal triangle.

Consider the question below.
I made a file for you to investigate
the case when the pedal point is the centroid of triangleABC.
Click here.
Do you find something?
I found only two things:
1. the ratio of the are of RST and
the are of ABC is between 0 and 0.25.
2. the pedal point is always inside
of RST.
You can check if these facts are true,
by clicking here.
I know these are very obvious and uninteresting...
When the pedal point is the incenter,
these two facts also can be found.
Well then, let's see the question below
which is about incenter.
You can investigate this case by clicking here.
In this case, as you see, the points
R, S, and T are the tangent points of the incircle of ABC.

When the pedal point is the orthocenter
(for the quesiton below),
the ratio of the area of RST and the
area of ABC is between 0 and 2.
Check this
clicking here.
In the case when the pedal point is
the circumcenter (for the question below),
the ratio of the triangles (RST and
ABC) is always 0.25.
Please look
at here!
The proof of the last fact is very
easy. It is because RT and BC are parallel, RS and CA are parallel,
and ST and AB are parallel. So, the areas of ART, RBS, TSC, and
RST are same.

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