tangent inequality in triangle

Let $\dfrac{\pi-A}4=x$ etc.

$\implies4(x+y+z)=3\pi-\pi\iff x+y+z=\dfrac\pi2$

Now as $\tan x,\tan y,\tan z$ are real,

$$(\tan x-\tan y)^2+(\tan y-\tan z)^2+(\tan z-\tan x)^2\ge0$$

$$\implies\tan^2x+\tan^2y+\tan^2z\ge\tan x\tan y+\tan y\tan z+\tan z\tan x$$

Finally $$\tan(x+y)=\tan\left(\dfrac\pi2-z\right)$$

$$\iff\dfrac{\tan x+\tan y}{1-\tan x\tan y}=\dfrac1{\tan z}$$

Simplify

Tags:

Inequality

Trigonometry

Triangles

Jensen Inequality

Geometric Inequalities

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