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 
quadrilateral DEBG is 
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/