The perpendicular distance from point P(x1, y1) to line ax + by + c = 0 is d = |ax1 + by1 + c|/sqrt($a^2$ + $b^2$). This is the single most important formula in the chapter. For the distance between parallel lines ax + by + c1 = 0 and ax + by + c2 = 0, d = |c1 - c2|/sqrt($a^2$ + $b^2$). Important: ensure the coefficients of x and y are identical before applying this formula. The distance from the origin is a special case with x1 = y1 = 0, giving |c|/sqrt($a^2$ + $b^2$). These formulas are used to find radii of inscribed/circumscribed circles, determine tangency conditions, and calculate areas of geometric figures. The signed distance (without absolute value) determines which side of the line a point lies on and is crucial for angle bisector identification.