Question

If the three points of contact of a circle inscribed in a triangle are joined, the angles of the resulting triangle:

• A. are always equal to ${60}^{\circ }$
• B. are always one obtuse angle and two unequal acute angles
• C. are always one obtuse angle and two equal acute angles
• D. are always acute angles
• E. are always unequal to each other.

Answer 1

Answer: (D)

are always acute angles

Consider a triangle ABC in which circle is inscribed

Let point of contact of the circle with triangle be D, E, F as shown in the figure

Let the angle at the vertices A, B, C be $\alpha ,\beta ,\gamma$respectively

As AF and AE are tangents to the circle from a same point A$\Rightarrow AF=AE\Rightarrow \angle AFE=\angle AEF=90-\frac{\alpha }{2}$

Similarly $\angle BFD=90-\frac{\beta }{2}$

$\therefore \angle DFE=\frac{\alpha +\beta }{2}$

As $\alpha +\beta <180\Rightarrow \frac{\alpha +\beta }{2}<90$

Similarly we can prove other interior angles are less than 90

Here all the interior angles are less than 90 $\Rightarrow$triangle is always acute

$\therefore$option 'D' is correct