In-a-triangle-the-bisector-of-the-side-c-is-perpendicular-to-side-b-Prove-that-2tanC-tanA-0-

Question Number 211672 by MATHEMATICSAM last updated on 15/Sep/24

In a triangle the bisector of the side c is

perpendicular to side b . Prove that

2 tanC + tanA = 0 .

Commented by Frix last updated on 16/Sep/24

If the bisector of a s i d e is perpendicular to

the other side \Rightarrow the angle between these

sides is either 0 ° or 180 ° .

The bisector of the a n g l e c cannot be

perpendicular to the side b because {it}^{'} s the

angle between the sides a and b .

Commented by Frix last updated on 16/Sep/24

δ = \sqrt{(a + b + c) (a + b - c) (a + c - b) (b + c - a)}

\tan α = \frac{δ}{b^{2} + c^{2} - a^{2}}

\tan β = \frac{δ}{a^{2} + c^{2} - b^{2}}

\tan γ = \frac{δ}{a^{2} + b^{2} - c^{2}}

2 \tan γ + \tan α = 0

\Leftrightarrow

a^{2} - 3 b^{2} - c^{2} = 0

Let c = 1 \Rightarrow a = \sqrt{3 b^{2} + 1} \land δ = 2 b \sqrt{1 - b^{2}}

\Rightarrow 0 < b < 1

\tan α = - \frac{\sqrt{1 - b^{2}}}{b}

\tan β = \frac{b \sqrt{1 - b^{2}}}{b^{2} + 1}

\tan γ = \frac{\sqrt{1 - b^{2}}}{2 b}

You can draw any of these to see {what}^{'} s

perpendicular or not \dots

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