Question

ABC is a triangle. If D is a point in the plane of the triangle such that the perpendicular distance from D to the three sides of the triangle are all equal, then there exist(s):

• A. just one such point as $D$
• B. three such point as $D$
• C. four such points as $D$
• D. none of the above

Answer 1

The correct option is A just one such point as $D$

The point D is equidistant from all the sides of the triangle .
Therefore it is the centre of the incircle of the triangle.
Let P be another point which is equidistant from the three sides of the triangle.
Therefore P is the centre of the incircle of the triangle.
So both P and D are the centre of the same circle.
This is not possible.
Therefore D is just one and only one such point.