THEOREMS IN ELEMENTARY GEOMETRY
1) Smarandache Concurrent Lines
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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)
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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
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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
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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.