A Generalization of Certain Remarkable Points of the Triangle Geometry

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A Generalization of Certain Remarkable Points of the Triangle Geometry Prof. Claudiu Coandă – National College “Carol I”, Craiova, Romania Prof. Florentin Smarandache – University of New Mexico, Gallup, U.S.A. Prof. Ion Pătrașcu – National College “Fraţii Buzeşti”, Craiova, Romania In this article we prove a theorem that will generalize the concurrence theorems that are leading to the Franke’s point, Kariya’s point, and to other remarkable points from the triangle geometry. Theorem 1: Let P (α , β , γ ) and A ', B ', C ' its projections on the sides BC , CA respectively AB of the triangle ABC . JJJJG JJJJG JJJG JJJJG JJJG JJJJG We consider the points A ", B ", C " such that PA " = k PA ' , PB " = k PB ' , PC " = k PC ' , where k ∈ R * . Also we suppose that AA ', BB ', CC ' are concurrent. Then the lines AA ", BB ", CC " are concurrent if and only if are satisfied simultaneously the following conditions: α β γ ⎛β ⎞ ⎛γ ⎞ ⎛α ⎞ αβ c ⎜ cos A − cos B ⎟ + βγ a ⎜ cos B − cos C ⎟ + γα b ⎜ cos C − cos A ⎟ = 0 a b c ⎝b ⎠ ⎝c ⎠ ⎝a ⎠ 2 2 2 α β γ α ⎛γ ⎞ β ⎛α ⎞ γ ⎛β ⎞ cos A ⎜ cos B − cos C ⎟ + 2 cos B ⎜ cos C − cos A ⎟ + 2 cos C ⎜ cos A − cos B ⎟ = 0 2 a b c a ⎝c ⎠ b ⎝a ⎠ c ⎝b ⎠ Proof: We find that α ⎛ α ⎞ A ' ⎜ 0, 2 ( a 2 + b 2 − c 2 ) + β , a 2 − b2 + c2 ) + γ ⎟ 2 ( 2a ⎝ 2a ⎠ JJJJG JJJG JG JG JJG ⎤ α α ⎡ PA " = k PA ' = k ⎢ −α rA + 2 ( a 2 + b 2 − c 2 ) rB + 2 ( a 2 − b 2 + c 2 ) rC ⎥ 2a 2a ⎣ ⎦ JJJJG JG JG JJG PA " = (α "− α ) rA + ( β "− β ) rB + ( γ "− γ ) rC We have:

⎧ ⎪α "− α = −kα ⎪ kα 2 ⎪ 2 2 ⎨ β "− β = 2 ( a + b − c ) , 2a ⎪ k α 2 2 2 ⎪ ⎪⎩γ "− γ = 2a 2 ( a − b + c ) Therefore:

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