The bisector of a regular triangle is 21. What is the radius of the circle inscribed in this triangle?

1. Vertices of triangle A, B, C. BH – bisector.

2. The bisector of a regular triangle also performs the functions of height and median.

Therefore, AH = 1/2 AC = 1/2 AB, and the triangle ABH is rectangular.

3. BH² = AB² – AH² (by the Pythagorean theorem).

BH² = AB² – AH² = 4AH² – AH² = 3AH².

AH² = 21² / 3 = 7 x 7 x 3.

AH = √7 x 7 x 3 = 7√3 cm.

AC = 7√3 x 2 = 14√3 cm.

4. The radius of a circle inscribed in a triangle (R) is calculated by the formula:

R = AC / 2√3 = 14√3 / 2√3 = 7 cm.

Answer: the radius of the circle inscribed in the ABC triangle is 7 cm.