FOM: conics
Matthew Frank
mfrank at math.uchicago.edu
Wed Feb 28 23:31:54 EST 2001
Neil--
> Can any fomer tell me who first proved that a tangent to an
> ellipse makes equal angles with the lines drawn from the foci
> to the point of contact of the tangent with the ellipsis? And
> where is the most *elementary* proof to be found?
I can't help with references off-hand, but here's an elementary proof:
Let F and G be the foci of the ellipse.
Let P be a point on the ellipse.
Let APB be the tangent to the ellipse at P.
Then of all points P' on AB, P is the one which minimizes the distance FP'
+ P'G (because all the other points on the line are outside the ellipse).
Let H be the reflection of G about the line AB.
Then for any point P' on AB, P'G = P'H.
So P is also the point on AB which minimizes the distance FP' + P'H.
Hence P is at the intersection of FH with AB.
So angle APF = angle HPB = angle BPG.
QED.
--Matt
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