вписанная окружность
геометрия на плоскости
доказательство
кривая второго порядка
симметрия
точка касания
фокальный радиус
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эллипс
AYLANA VA ELLIPSNING TEGISH NUQTALARINI ANIQLASH
Nashr sanasi
25.06.2026
Jurnal
Ta'lim ufqlari
Nashr
"Ta'lim ufqlari" ilmiy-uslubiy jurnali 2026-yil 1-son
Sahifalar
71-73
Mualliflar
Annotatsiya
Ushbu maqolada ellipsga ichki chizilgan ikkita aylana va ularning ellips bilan tegish nuqtalarini birlashtiruvchi to‘g‘ri chiziqlar haqidagi teoremanining isboti taqdim etiladi. Teorema shuni ta’kidlaydiki, bunday har qanday to‘g‘ri chiziq ikkala aylanadan teng uzunlikdagi vatalalar kesib oladi. Isbot ellipsning fokal xossalari, ichki chizilgan aylana markaziga tushirilgan perpendikulyar va vatalani hisoblashning klassik formulasidan foydalanadi. Natija ellips va aylana o‘rtasidagi simmetriyaning nozik xususiyatini ochib beradi.
Kalit so‘zlar
ellips
fokal radius
ichki chizilgan aylana
ikkinchi tartibli egri chiziq
isbotlash
simmetriya
tegish nuqtasi
tekislik geometriyasi
vatala
Boshqa tillardagi variantlar
This article presents a proof of the theorem concerning two circles inscribed in an ellipse and the lines connecting their tangent points with the ellipse. The theorem states that any such line cuts equal chords from both inscribed circles. The proof employs the focal properties of the ellipse, perpendiculars dropped from the centres of the inscribed circles to the chord line, and the classical chord-length formula. The result reveals a subtle symmetry property between an ellipse and its inscribed circles.
chord
ellipse
focal radius
inscribed circle
plane geometry
proof
second-order curve
symmetry
tangent point
Foydalanilgan adabiyotlar
1. Prasolov V.V., Tikhomirov V.M. Geometriya. – Moskva: MSNMO, 2007. – 336 b.
2. Efimov N.V. Analitik geometriya kursi. – Toshkent: O‘qituvchi, 1985. – 312 b.
3. Berger M. Geometry I. – Berlin: Springer, 2009. – 427 p.
4. Coxeter H.S.M. Introduction to Geometry. – New York: Wiley, 1969. – 469 p.
5. Uspensky V.A. Ellips, giperbola, parabola. – Moskva: Nauka, 1973. – 96 b.