Образователни технологии
A NEW PROOF OF THE FEUERBACH THEOREM
Резюме. By using the Paskalev-Tchobanov distance formula, we give a new proof of the famous Feuerbach theorem.
Ключови думи: Paskalev-Tchobanov distance formula; Feuerbach theorem
The famous Feuerbach theorem states (See e.g. (Weisstein)):
Theorem 1. The nine-point circle of any triangle is tangent internally to the incircle and tangent externally to the three excircles.
Proof. Given triangle \(A B C\) with side-lengths \(a=B C, b=C A, c=A B\). Denote
by \(-s=\tfrac{a+b+c}{2}\) the semiperimeter of triangle \(A B C\).
– \(\Delta\) the area of triangle \(A B C\).
\(-r\) the radius of the Incircle.
\(-R\) the radius of the Circumcircle.
\(-r_{N}\) the radius of the Nine-Point Circle.
– \(r_{a}\) the radius of the A-Excircle.
\(-r_{b}\) the radius of the B-Excircle.
\(-r_{c}\) the radius of the C-Excircle.
We use the following well-known formula e:
\[ \begin{gathered} \Delta=\sqrt{s(s-a)(s-b)(s-c)}, \quad r=\tfrac{\Delta}{s} . \quad R=\tfrac{a b c}{4 \Delta}, \quad R_{N}=\tfrac{R}{2}, \quad r_{a}=\tfrac{\Delta}{s-a} \\ r_{b}=\tfrac{\Delta}{s-b}, \quad r_{c}=\tfrac{\Delta}{s-c} . \end{gathered} \]
Denote
\[ \begin{gathered} E=a^{3}+b^{3}+c^{3}+3 a b c-a^{2} b-a b^{2}-a^{2} c-a c^{2}-b^{2} c-b c^{2} \\ E_{1}=a^{3}-b^{3}-c^{3}+3 a b c+a^{2} b-a b^{2}+a^{2} c-a c^{2}+b^{2} c+b c^{2} \\ E_{2}=-a^{3}+b^{3}-c^{3}+3 a b c-a^{2} b+a b^{2}+a^{2} c+a c^{2}+b^{2} c-b c^{2} \\ E_{3}=-a^{3}-b^{3}+c^{3}+3 a b c+a^{2} b+a b^{2}-a^{2} c+a c^{2}-b^{2} c+b c^{2} \end{gathered} \]
Note that the above forms \(E_{1}, E_{2}, E_{3}\) are used in (Grozdev, Okumura & Dekov, 2018). By using the above forms and formulae, we see easily, that
\[ r_{N}-r=\tfrac{E}{8 \Delta}, \quad r_{N}+r_{a}=\tfrac{E_{1}}{8 \Delta}, \quad r_{N}+r_{b}=\tfrac{E_{2}}{8 \Delta}, \quad r_{N}+r_{c}=\tfrac{E_{3}}{8 \Delta} . \]
The None-Point Circle is tangent internally to the Incircle iff \(N I=r_{N}-r\) and it is tangent externally to the A-Excircle iff \(N J_{a}=r_{N}+r_{a}\). Similar results hold for the B-Excircle and C-Excircle. The barycentric coordinates of the centers of the above circles are as follows (See e.g. (Grozdev & Dekov, 2016), (Weisstein)):
\[ \begin{aligned} & N=a^{2} b^{2}+a^{2} c^{2}+2 b^{2} c^{2}-b^{4}-c^{4}:: \\ & I=(a, b, c), \quad J_{a}=(-a, b, c), \quad J_{b}=(a,-b, c), \quad J_{c}=(a, b,-c) \end{aligned} \]
Now we can use the Paskalev-Tchobanov distance formula (See (Paskalev & Tchobanov, 1985), or formula (9) in (Grozdev & Dekov, 2016)). By using the Paskalev-Tchobanov distance formula we obtain \(N I=\tfrac{E}{8 \Delta}\). Hence \(r_{N}-r=N I\). Similarly, we see that \(r_{N}+r_{a}=N J_{a}, r_{N}+r_{b}=N J_{b}\) a and \(r_{N}+r_{c}=N J_{c}\). This completes the proof.
REFERENCES
Grozdev, S. & Dekov, D. (2016). Barycentric Coordinates: Formula Sheet, International Journal of Computer Discovered Mathematics, vol.1, no 2, 75 – 82, available at the web.
Grozdev, S., Okumura H. & Dekov D. (2018) A Note on the Feuerbach triangle, Mathematical Gazette, vol.102, no.553, 135 – 138.
Paskalev, G. & Tchobanov I. (1985). Remarkable points in triangle. Sofia: Narodna Prosveta (in Bulgarian).
Weisstein, E. W. MathWorld – A Wolfram Web Resource, available at the web.