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.

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.

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.

return