Elem-geo

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 --------------------------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 Podaire 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: a'b' + b'c' + c'a' <= 1/4 (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.

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 = 2AB.BC.CD + (AB

2 + BC

2 3 3 3 + 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.