The base of a triangle and the ratio of the tangents of half angle on the base are given. Prove that the locus of the vertex of the triangle is a hyperbola.
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Solution
Let BC be the base of the triangle and A is the vertex.
tan(B2)tan(C2)=√(s−a)(s−c)s(s−b)√(s−a)(s−b)s(s−c)=s−cs−b=2s−2c2s−2b=a+b−ca+c−b=k (let it be.)
By componendo-diviendo, we have
⇒k−1k+1=b−ca
⇒b−c=a(k−1k+1)=constant
⇒AC−AB=constant
Therefore, the difference of distance of point A from the two foci B and C is constant.