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### Simson Lines

```
Date: 06/07/2001 at 02:54:39
From: Gabriel Kreindler
Subject: Simson's lines

If we have two triangles inscribed in the same circle, is it true that
for any point P on the circle the two Simson's lines for the point P
for the two triangles make between them the same angle?

If the answer is "yes", then how can you prove it?

A thousand thanks,
Kappa
```

```
Date: 06/07/2001 at 06:55:28
From: Doctor Floor
Subject: Re: Simson's lines

Hi, Kappa,

Thanks for writing.

The key for the solution of this problem is found by considering what
happens with one triangle and a moving point on its circumcircle.

Look at the Dr. Math archives page on the Simson line:

http://mathforum.org/dr.math/problems/furman.04.19.99.html

and in particular one of its figures:

ABC is the triangle, D a point on its circumcircle and the three
points EFG form the Simson line. It is easy to see that the
cyclic, because angles E and G are right angles. From this we see

<CGE = 90 deg - <DGE = 90 deg - <DBE = 90 deg - <DBA

(because <DBE and <DGE are inscribed angles on the same arc).

Now we see that when D moves through an arc of a certain measure t,
then <DBA increases by t (inscribed angle) and thus <CGE decreases by
t, and thus the angle that the Simson line and BC make decreases by t.

So the Simson line 'rotates' (it does not rotate about a fixed point)
with the same 'speed' as point D, but in opposite direction.

Now it is easy to see that when we have two triangles, the Simson
lines of a moving point D both move with the same speed and in the
same direction. So their angle stays fixed.

If you need more help, just write back.

Best regards,
- Doctor Floor, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Conic Sections/Circles
High School Geometry
High School Triangles and Other Polygons

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