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Owner confidence rating: High Entry average rating: No information on entry rating
Simson's line (Theorem)

Let $ABC$ a triangle and $P$ a point on its circumcircle (other than $A,B,C$). Then the feet of the perpendiculars drawn from P to the sides $AB,BC,CA$ (or their prolongations) are collinear.

\includegraphics{simson.eps}

In the picture, the line passing through $U,V,W$ is a Simson line for $\triangle ABC$.

An interesting result form the realm of analytic geometry states that the envelope formed by Simson's lines when P varies is a circular hypocycloid of three points.

\includegraphics{simson-env}



"Simson's line" is owned by drini. [ owner history (1) ]
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See Also: circumcircle, triangle


Attachments:
proof of Simson's line (Proof) by giri
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Cross-references: circular, envelope, analytic geometry, passing through, line, collinear, sides, perpendiculars, circumcircle, point, triangle

This is version 14 of Simson's line, born on 2002-02-20, modified 2005-02-05.
Object id is 2277, canonical name is SimsonsLine.
Accessed 4972 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
Pending Errata and Addenda
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