Level 14: Construct the in-circle of a triangle:

Hint: The center of the in-circle must have same distance to the triangle lines. In particular, two of the three lines?

Solution: Consider the set of all points that is of the same distance to both AB and AC, what is it? It is the angle bisector of angle BAC! As the hint stated, the center must lie on the angle bisectors, so the angle bisectors intersects and that is the center of the in-circle. We only need two angle bisectors (of angle ABC and angle BAC) to find the center of the in-circle. To find the radius of the in-circle, note that the side must be perpendicular to the radius (as it is tangent), so we construct the perpendicular through the center to AB, that give us the intersection point F and that is the radius.