Feuerbachova kružnica i Simsonov pravac

Abstract

U ovom radu proučavali smo kružnicu devet točaka (Feuerbachovu kružnicu), Feuerbachov teorem i Simsonov pravac. Za svaki trokut $ABC$, polovišta stranica $A', B',C'$, nožišta visina $D, E, F$ te točke $M, N, P$ koje su polovišta dužina $\overline{AH}, \overline{BH}, \overline{CH}$, pri čemu je $H$ ortocentar danog trokuta, leže na kružnici $k_9$ koju nazivamo kružnicom devet točaka ili Feuerbachovom kružnicom. Središte te kružnice je polovište dužine $\bar{HO}$, gdje je $O$ središte opisane kružnice trokuta $ABC$, a njen polumjer je jednak polovini polumjera opisane kružnice. Prema Feuerbachovom teoremu, kružnica devet točaka dodiruje trokutu $ABC$ upisanu i sve tri pripisane kružnice, te vrijedi: \begin{align*} |O_9U| = \frac{R}{2}-r, &\: |O_9U_a| = \frac{R}{2}+r_a,\\ |O_9U_b|= \frac{R}{2}+r_b, &\: |O_9U_c|= \frac{R}{2}+r_c \end{align*} pri čemu je točka $O_9$ središte kružnice devet točaka, $R$ polumjer trokutu opisane kružnice, $U, U_a, U_b, U_c$središta upisane odnosno pripisanih kružnica te $r, r_a, r_b, r_c$ redom njihovi polumjeri. Simsonov pravac je pravac na kojem leže nožišta okomica spuštenih iz neke točke opisane kružnice trokuta na stranice tog trokuta. Ako je $H$ ortocentar danog trokuta, Simsonov pravac neke točke $P$ koja leži na opisanoj kružnici tog trokuta siječe dužinu $\overline{PH}$ u njenom polovištu i to polovište leži na Feuerbachovoj kružnici tog trokuta. Nadalje, Simsonovi pravci dijametralno suprotnih točaka opisane kružnice danog trokuta su međusobno okomiti i sijeku se na Feuerbachovoj kružnici tog trokuta.In this thesis we studied the nine-point circle (Feuerbach’s circle), Feuerbach’s theorem and the Simson line. For each triangle $ABC$, the midpoints $A', B',C'$ of its sides, the feet $D, E, F$ of its altitudes, and points $M, N, P$ which are the midpoints of line segments $\overline{AH}, \overline{BH}, \overline{CH}$, where $H$ is the orthocentre of a given triangle, lie on a circle which is called the nine-point circle or Feuerbach’s circle. The centre of that circle is the midpoint of line segment $\overline{HO}$, where $O$ is the centre of the circumscribed circle of the triangle $ABC$, and its radius is half of the radius of the circumscribed circle. According to Feuerbach’s theorem, the nine-point circle touches the inscribed circle and all three escribed circles of triangle, and the following holds true: \begin{align*} |O_9U| = \frac{R}{2}-r, &\: |O_9U_a| = \frac{R}{2}+r_a,\\ |O_9U_b|= \frac{R}{2}+r_b, &\: |O_9U_c|= \frac{R}{2}+r_c \end{align*} where $O_9$ is the centre of the nine-point circle, $R$ is the radius of the triangle’s inscribed circle, $U, U_a, U_b, U_c$ are the centres of inscribed and escribed circles and $r, r_a, r_b, r_c$ are their radii, respectively. The Simson line is a line through the feet of perpendiculars from a point of the triangle’s circumscribed circle to the sides of that triangle. If $H$ is the orthocentre of a given triangle, the Simson line of a point $P$ that lies on that triangle’s circumscribed circle intersects the line segment $\overline{PH}$ in its midpoint and that midpoint lies on Feuerbach’s circle of that triangle. Furthermore, the Simson lines of diametrically opposite points of the circumscribed circle of given triangle are perpendicular and they intersect on that triangle’s Feuerbach’s circl