in every triangle we can inscribe a circle I am trying to show that in every triangle we can inscribe a circle. I reduced it to following: in every triangle there must be a point in the interior, such that there are three points on the triangle (each one on one side) such that the distance from any one of them to the point in the interior is the same. How can I prove this?

**Hint:** the centre of the circle should be equidistant of the sides of the triangle.

**Hint:** The bisector of an angle is the set of points equidistant of the sides of the angle.

**Hint:** The distance between a point and a line is the length of the perpendicular segment from that point to that line.