Another proof of a theorem relative to the orthological triangles

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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


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