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Nagel Point: What Is It About?
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Copyright © 1996-2009 Alexander Bogomolny
The applet may suggest the following statement:
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Let E, D, F be the points of tangency of the excircles of |
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The proof relies on Ceva's theorem.
First, let CC denote the excircle of 
ABC opposite the vertex C. CC is tangent to the side AB and the side lines AC and BC in points F, T and U. Since, CT and CU are the two tangents from C to CC, they are equal in length:
| (1) | CT = CU. |
For a similar reason
| (2) | AT = AF, and |
| (3) | BU = BF. |
(1)-(3) immediately imply that F is a perimeter splitter in the sense that
| AC + AF = BC + BF. |
If a, b, c are the side lengths of the ABC and p its semiperimeter in a standard way, then this fact could be rewritten as
| b + AF = a + BF = p, |
so that
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AF = p - b BF = p - a. |
Similarly, we obtain additional four identities:
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AE = p - c CE = p - a CD = p - b BD = p - c. |
Ceva's identity is then verified directly:
| AE/CE·CD/BD·BF/AF = (p-c)/(p-a)·(p-b)/(p-c)·(p-a)/(p-b) = 1. |
(A curious fact is that these same points D, E, F arise in a related, yet quite a different construction.)
Copyright © 1996-2009 Alexander Bogomolny
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