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Afine transformacije djeluju na jednakostranični trokut i njemu opisanu odnosno upisanu kružnicu na način da jednakostranični trokut preslikaju u općeniti trokut i tom trokutu upisanu elipsu odnosno opisanu elipsu. Upravo te elipse nazivaju se Steinerovom upisanom i opisanom elipsom trokuta. Jedinstvene su sa svojstvima da upisana elipsa dodiruje stranice trokuta u njegovim polovištima, a da opisana elipsa ima središte u težištu trokuta. Nakon dokaza postojanja i jedinstvenosti Steinerovih elipsi, pokazano je da su te elipse povezane homotetijom s koeficijentom 2. Steinerova opisana elipsa ima najmanju površinu od svih opisanih elipsa, a Steinerova upisana elipsa ima najveću površinu među svim upisanim elipsama trokuta. Nadalje, dokazan je Siebeckov teorem i njegova veza sa Steinerovim elipsama. Ako su nultočke kubnog polinoma s kompleksnim koeficijentima vrhovi trokuta u kompleksnoj ravnini, nultočke derivacije tog polinoma su fokusi Steinerove upisane elipse tog trokuta. Fokusi Steinerove opisane elipse imaju svojstvo da su to ekvicevijane točke trokuta, što znači da su to točke koje određuju tri cevijane jednakih duljina. Ovo svojstvo dokazano je analitičkom i sintetičkom metodom u zadnjem dijelu rada. Affine transformations map an equilateral triangle and its incircle resp. circumcircle onto a general triangle and its inscribed ellipse resp. circumscribed ellipse. These ellipses are called Steiner’s inellipse and circumellipse. They are unique with the property that the inellipse is tangent to the sides at their midpoints, and that the circumellipse is centered at the center of gravity of the triangle. After establishing the existence and uniqueness of Steiner’s ellipses, it is shown that these ellipses are related by a homothety with coefficient 2. Steiner’s circumellipse has the smallest area of all circumscribed ellipses, and the Steiner inellipse has the largest area among all inscribed ellipses. Furthermore, Siebeck’s theorem and its connection with Steiner ellipses is proven. If zeros of a cubic polynomial with complex coefficients determine the vertices of a triangle in the complex plane, zeros of the derivative of that polynomial are the foci of Steiner’s inellipse to that triangle. The foci of Steiner’s circumellipse have the property that they are equicevian points of a triangle, which means that these are the points defining three cevians of equal length. This property is proven by analytical and synthetic methods in the last part of the paper.

U ovom diplomskom radu su proučavani odnosi između položaja nultočaka polinoma p i nultočaka derivacije p' tog polinoma. Najpoznatiji teorem iz tog područja je Gauss - Lucasov teorem, koji kaže da se nultočke derivacije p' nalaze u konveksnoj ljusci skupa nultočaka od p. Također smo pokazali neke srodne rezultate, poput Siebeckova teorema. U njemu je dana geometrijska veza između nultočaka kubnog polinoma i nultočaka njegove derivacije. Teorem tvrdi da ako su nultočke od p nekolinearne točke kompleksne ravnine tada su nultočke derivacije p' žarišta jednoznačno određene elipse upisane trokutu nultočaka koja dodiruje stranice trokuta u polovištima. U dokazima su korištene afine transformacije, pa smo naveli i dokazali neke činjenice o afinim i linearnim transformacijama, posebno u kompleksnoj ravnini. The relationship between the location of the zeros of a complex polynomial p and those of its derivative has been studied in this graduate thesis. The best-known theorem in this area is the Gauss-Lucas Theorem, that the zeros of p' lie in the convex hull of the zeros of p. A special case, Siebeck's theorem is also interesting. That theorem describes geometric connection between the zeros of a cubic polynomial p with complex coefficients and the zeros of the polynomial's derivate p'. To wit, if the zeros of p are noncollinear points z_1, z_2 and z_3 in the complex plane, the zeros of p' are the foci of the unique ellipse inscribed in triangle z_1z_2z_3 and tangent to the sides at their midpoints. Our proofs makes use of affine transforma tions, so we recall and prove the necessary facts about linear and affine transformations, especially in the complex plane.