THEOREMS IN ELEMENTARY GEOMETRY 1) Smarandache Concurrent Lines ---------------------------- If a polygon with n sides (n >= 4) is circumscribed to a circle, then there are at least three concurrent lines among the polygon's diagonals and the lines which join tangential points of two non-adjacent sides. (This generalizes a geometric theorem of Newton.) Reference: F. Smarandache, "Problemes avec and sans problemes!" (French: Problems with and without ... Problems!), Ed. Somipress, Fes, Morocco, 1983, Problem & Solution # 5.36, p. 54. 2) Smarandache Cevians Theorem (I) ------------------------------- Let AA', BB', CC' be three concurrent cevians (lines) in the point P in the triangle ABC. Then: PA/PA' + PB/PB' + PC/PC' >= 6, and PA PB PC BA CB AC ---- . ---- . ---- = ---- . ---- . ---- >= 8. PA' PB' PC' BA' CB' AC' Reference: F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress, Fes, Morocco, 1983, Problems & Solutions # 5.37, p. 55, # 5.40, p. 58. 3) Smarandache Orthic Theorem --------------------------- Let AA', BB', CC' be the altitudes (heights) of the triangle ABC. Thus A'B'C' is the podaire triangle of the triangle ABC. Note AB = c, BC = a, CA = b, and A'B' = c', B'C' = a', C'A' = b'. Then: 4(a'b' + b'c' + c'a') <= a^2 + b^2 + c^2 Reference: F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress, Fes, Morocco, 1983, Problem & Solution # 5.41, p. 59. C. Barbu, Teorema lui Smarandache, in his book “Teoreme fundamentale din geometria triunghiului”, Chapter II, Section II.57, p. 337, Editura Unique, Bac?u, 2008. 4) Generalization of the Bisector Theorem -------------------------------------- Let AM be a Cevian of the triangle ABC which forms the angles A1 and A2 with the sides AB and AC respectively. Then: BA BM sin A2 ---- = ----.-------- CA CM sin A1 Reference: F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev University Press, Kishinev, Problem 61, pp. 41-42, 1997. 5) Generalization of the Altitude Theorem -------------------------------------- Let AD be the altitude of the triangle ABC which forms the angles A1 and A2 with the sides AB and AC respectively. Then: 2 AD = BD.DC.cot A1.cot A2 Reference: F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev University Press, Kishinev, Problem 62, pp. 42-43, 1997. 6) Collinear Points Theorem ------------------------ Let A, B, C, D be collinear points and O a point not on their line. Then: 2 2 2 2 (OA - OC )BD + (OD - OB )AC = 2 2 2 3 3 3 = 2AB.BC.CD + (AB + BC + CD )AD - (AB + BC + CD ) Reference: F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev University Press, Kishinev, Problem 82, p. 61, 1997. 7) Median Point Theorem -------------------- Let P be a point on the median AA' of the triangle ABC. One notes by B' and C' the intersections of BP with AC and of CP with AB respectively. Then: a) B'C' is parallel to BC. b) In the case when AA' is not a median, let A'' be the intersection of B'C' with BC. Then A' and A'' divide BC in an anharmonic rapport. Reference: F. Smarandache, "Proposed Problems of Mathematics", Vol. II, Kishinev University Press, Kishinev, Problem 81, p. 60, 1997.