Recall this theorem:[br][br][i]If a point lies on the bisector of an angle, then it is equidistant from the sides (rays) of that angle. [br][br][/i]From the screencast:[br]Since [i]A[/i] lies on the bisector of [math]\angle EDF[/math], it is equidistant from [math]\overline{DE}[/math] and [math]\overline{DF}[/math]. [br]Since [i]A[/i] lies on the bisector of [math]\angle DEF[/math], it is equidistant from [math]\overline{DE}[/math] and [math]\overline{EF}[/math]. [br]Since [i]A[/i] lies on the bisector of [math]\angle EFD[/math], it is equidistant from [math]\overline{DF}[/math] and [math]\overline{EF}[/math]. [br][br]Thus, [i]A[/i] is equidistant from all 3 sides of the triangle. Because of this, [i]A[/i] serves as the center of the one circle we can inscribe inside the triangle. This inscribed circle is tangent to the triangle's 3 sides and is the LARGEST circle we can draw inside the triangle. [br][br]Point [i]A[/i] is called the incenter of the triangle. [br]