15
2
2
2 2 A A B − A B −
2
2
2
2 a Ca C B − + +
=
2
2
2
2
2 a Ca C B + + +
2
A
2
C
2
2
2 a Ca A − +
2
A
=
2
C
2
2
2 a Ca A + +
2
A
C
2
a A −
(
)
A
=
C
2
a A +
(
)
A
C
a A −
A
=
C
a A +
A
1 2 FT FG F T F H = 1 2
o
and
FGT F HT ∠ = ∠ =
90
1
2
So, Triangle F 1 GT is similar to Triangle F 2 HT. (3)
⇒ ∠ = ∠ ⇒ =
1 2 FTG F TH β α
All considered, the proposition is now proved!
This proof shows a property of an ellipse, which is of great use in real life. We can find it in modern medical equipments, where people use it to concentrate high energy to break the stone in human bodies. It also has connection with sonar system used to locate submarines. Exciting, isn’t it? Finally, ellipses have a lot of properties. Other properties can all be proved mathematically. This is, in my opinion, why mathematics is of great fun and challenge. You may find the other properties through various ways and have a try to prove it on your own! Thank you!
(1) http://en.wikipedia.org/wiki/Implicit_function (2) http://en.wikipedia.org/wiki/Distance_from_a_point_to_a_line (3) http://mathforum.org/library/drmath/view/57674.html (prove triangles similar
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