Properties of the Adjoin Circles to a Triangle

In this article we'll present the properties of the radicale axes and the adjoin circles of a triangle. Definition 1 Given a triangle ABC, we call the circle that passes through the vertexes C, A and it is tangent in the point A to the side AB, that it is an adjoin circle to the given triangle. Observations a) We note the circle from the above definition CĀ. b) To a triangle, in general, there are corresponding 6 different adjoin circles. If the given triangle is isosceles, it will have 5 different adjoin circles, and if the triangle is equilateral, there will be 3 different adjoin circles associated to it. i). The adjoin circles , A C , B A , C B of a random triangle ABC have a common point Ω with the property: AB BC CA ∠Ω ≡ ∠Ω ≡ ∠Ω ii). The adjoin circles , C A , B C , A B of a random triangle ABC have a common point ' Ω with the property: ' ' ' AC BA CB ∠Ω ≡ ∠Ω ≡ ∠Ω Proof: i). Let Ω a second point of intersection of the circles A C and , B A (see fig.1). We have: CA AB şi AB BC ∠Ω ≡ ∠Ω ∠Ω ≡ ∠Ω. Indeed, the first angles have as measure the half of the measure of the cord AΩ , and those from the second congruence have as measure half from the measure of the cord BΩ. We obtain that: AB BC CA ∠Ω ≡ ∠Ω ≡ ∠Ω. The relation BC CA ∠Ω ≡ ∠Ω show that the circumscribed circle to the triangle C BΩ is tangent in C to the side AC and it is, therefore, the adjoin circle C B. Observations: a). Similarly it can be proved ii). b). The point Ω is called the first point of Brocard (1) , and ' Ω is called the second point of Brocard.