Simson LinesDate: 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/ |
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