Multiple choice

In a triangle ABC, points D and E are taken on sides BC such that $DB=DE=EC$. If $\angle ADE=\angle AED=\theta $, then

  1. $\tan { \theta } =3\tan { B } $
  2. $\tan { \theta } =3\tan { C } $
  3. $\tan { A } =\cfrac { 6\tan { \theta } }{ \tan ^{ 2 }{ \theta } } $
  4. $9\cot ^{ 2 }{ \cfrac { A }{ 2 } } =\tan ^{ 2 }{ \theta }$
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A Correct answer
Explanation

Given DB=DE=EC, let each segment be x. In triangle ADE, using the sine rule or trigonometric ratios, one can derive the relationship between the angles. The correct identity derived from the geometry is tan(theta) = 3 tan(B).

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