Question

In ABC, the angle bisectors meet at point D. Point E is on AC, DE and is perpendicular to AC . Point F is the location where the perpendicular bisectors of the sides of the triangle meet. What is the radius of the largest circle that can fit inside ABC?

Answer 1

Point D is the center of the incircle ( inscribed circle) of the triangle.
Line AC is tangent to the incircle. If DE is perpendicular to AC, then the radius of the largest circle that can fit inside triangle ABD is: DE.

Answer 2

Answer:

The radius is DE.

Step-by-step explanation:

Given that

In ABC, the angle bisectors meet at point D. Point E is on AC, DE and is perpendicular to AC . Point F is the location where the perpendicular bisectors of the sides of the triangle meet.

we have to find the radius of the largest circle that can fit inside ABC.

The incenter of the triangle is the point formed by the intersection of the triangle's 3 angle bisectors.  

Here, the angle bisectors meet at point D therefore the incenter is point D and the side AC is the tangent to the circle that fits inside ABC.

Hence, the radius is DE.