Another proof of a theorem relative to the orthological triangles Ion Pătraşcu – National College “Fraţii Buzeşti, Craiova, Romania Florentin Smarandache – New Mexico University, U.S.A. In [1] we proved, using barycentric coordinates, the following theorem: Theorem: (generalization of the C. Coşniţă theorem) If P is a point in the triangle’s ABC plane, which is not on the circumscribed triangle, A ' B ' C ' is its pedal triangle and A1 , B1 , C1 three points such that uuuuruuur uuuuruuur uuuuruuuur PA '⋅PA1 = PB '⋅PB1 = PC '⋅PC1 = k , k ∈ R* , then the lines AA1 , BB1 , CC1 are concurrent. Bellow, will prove, using this theorem, the following: Theorem If the triangles ABC and A1 B1C1 are orthological and their orthological centers coincide, then the lines AA1 , BB1 , CC1 are concurrent (the triangles ABC and A1 B1C1 are homological). Proof: Let O be the unique orthological center of the triangles ABC and A1 B1C1 and A C1
X1 Y3
B
Y2
O
X2
X3
Y1
{ X1} = AO I B1C1 { X 2 } = BO I A1C1 { X 3 } = CO I A1B1 We denote
B1
A1
C
{Y1} = OA1 I BC {Y2 } = OB1 I AC {Y3 } = OC1 I AB We observe that Therefore:
OAY3 = OC1 X 1 (angles with perpendicular sides).
OY3 OA , OX 1 sin OC1 X 1 = OC1 sin OAY3 =
then OX 1 ⋅ OA = OY3 ⋅ OC1
(1)
Also OC1 X 2 = OBY3 therefore sin OC1 X 2 = sin OBY3 =
OX 2 OC1
OY3 OB
and consequently: OX 2 ⋅ OB = OY3 ⋅ OC1 (2) Following the same path: OX 2 OY sin OA1 X 2 = = sin OBY1 = 1 OC1 OB from which OX 2 ⋅ OB = OA1 ⋅ OY1 (3) Finally OX 3 OY sin OA1 X 3 = = sin OCY1 = 1 OA1 OC from which: OX 3 ⋅ OC = OA1 ⋅ OY1 (4) The relations (1), (2), (3), (4) lead to OX 1 ⋅ OA = OX 2 ⋅ OB = OX 3 ⋅ OC (5) From (5) using the Coşniţă’s generalized theorem, it results that A1 A, B1 B, C1C are concurrent. Observation: If we denote P the homology center of the triangles ABC and A1 B1C1 and d is the intersection of their homology axes, them in conformity with the Sondat’s theorem, it results that OP ⊥ d .
References: [1] Ion Pătraşcu – Generalizarea teoremei lui Coşniţă – Recreaţii Matematice, An XII, nr. 2/2010, Iaşi, Romania [2] Florentin Smarandache – Multispace & Multistructure, Neutrosophic Transdisciliniarity, 100 Collected papers of science, Vol. IV, 800 p., NorthEuropean Scientific Publishers, Honka, Finland, 2010.