Straight Lines

Slope of the line passing through (x₁, y₁) and (x₂, y₂). Undefined when x₁ = x₂.

Slope in terms of the angle θ the line makes with the positive x-axis. θ ∈ [0°, 180°), θ ≠ 90°.

Three points (x₁,y₁), (x₂,y₂), (x₃,y₃) are collinear if and only if the slopes of any two pairs are equal.

Equivalent determinant condition for collinearity of three points.

m is the slope, c is the y-intercept. Every non-vertical line can be written in this form.

Line with slope m passing through (x₁, y₁).

Line passing through (x₁, y₁) and (x₂, y₂).

Line with x-intercept a and y-intercept b. Fails when the line passes through the origin.

p > 0 is the perpendicular distance from the origin to the line. α is the angle the perpendicular makes with the positive x-axis.

Most general equation of a straight line. Slope = −a/b (b ≠ 0), x-intercept = −c/a, y-intercept = −c/b.

Slope of ax + by + c = 0, valid when b ≠ 0.

Point at signed distance r from (x₁, y₁) along the line of inclination θ. r > 0 in the direction of increasing θ, r < 0 in the opposite direction.

Coordinates of the point at signed distance r from (x₁, y₁) along a line of inclination θ.

Two lines are parallel if and only if their slopes are equal (or both are vertical).

Two non-vertical lines are perpendicular if and only if the product of their slopes is −1.

Line through (x₁, y₁) parallel to ax + by + c = 0.

Line through (x₁, y₁) perpendicular to ax + by + c = 0.

Perpendicular distance from point (x₁, y₁) to the line ax + by + c = 0.

Distance between ax + by + c₁ = 0 and ax + by + c₂ = 0.

Foot of the perpendicular from (x₁, y₁) to the line ax + by + c = 0.

Image (reflection) of point (x₁, y₁) in the line ax + by + c = 0.

Acute angle θ between two lines with slopes m₁ and m₂. The obtuse angle is π − θ.

Acute angle between a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0.

Locus of points equidistant from both lines. The + sign gives one bisector, the − sign gives the other.

If a₁·a₂ + b₁·b₂ > 0, this bisector (+ sign, with c₁, c₂ same sign) contains the origin. Reverse if < 0.

Two points lie on the same side of ax + by + c = 0 if the expressions ax₁ + by₁ + c and ax₂ + by₂ + c have the same sign, and on opposite sides if signs differ.

Area of triangle with vertices (x₁,y₁), (x₂,y₂), (x₃,y₃).

Area of the triangle formed by three lines a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0, a₃x+b₃y+c₃=0.

Point of intersection of the medians. Divides each median in ratio 2:1 from vertex.

The altitude from A is perpendicular to BC. Orthocenter H is found by intersecting any two altitudes.

Circumcenter O is the intersection of perpendicular bisectors of the sides.

Where a, b, c are the lengths of the sides opposite to vertices A, B, C respectively.

Three lines a₁x+b₁y+c₁=0, a₂x+b₂y+c₂=0, a₃x+b₃y+c₃=0 are concurrent if and only if this determinant vanishes.

For any value of λ, this represents a line passing through the intersection of L₁: a₁x+b₁y+c₁=0 and L₂: a₂x+b₂y+c₂=0.

The fixed point of the family is the intersection of L₁ = 0 and L₂ = 0, independent of λ.

When the origin is shifted to (h, k), new coordinates (X, Y) relate to old by x = X + h, y = Y + k.

When axes are rotated by angle θ anticlockwise, new coordinates (X, Y) relate to old (x, y) by these equations.

Represents a pair of straight lines through the origin if and only if h² ≥ ab. The two lines are real and distinct (h²>ab), coincident (h²=ab), or imaginary (h²<ab).

For the pair ax²+2hxy+by²=0, the two slopes satisfy these Vieta-like relations.

Acute angle between the two lines represented by ax²+2hxy+by²=0.

The two lines in ax²+2hxy+by²=0 are perpendicular if and only if the sum of coefficients of x² and y² is zero.

The two lines in ax²+2hxy+by²=0 are coincident if and only if h²=ab.

Combined equation of the angle bisectors of the pair ax²+2hxy+by²=0. The bisectors are always perpendicular to each other.

Represents a pair of straight lines if and only if the discriminant Δ = 0.

i.e. abc + 2fgh − af² − bg² − ch² = 0. This is the necessary and sufficient condition.

Joint equation of lines joining the origin to the points of intersection of the conic ax²+2hxy+by²+2gx+2fy+c=0 and the line lx+my+n=0. The result is always a homogeneous second degree equation.

After homogenization, the chord of intersection subtends a right angle at the origin if and only if the sum of coefficients of x² and y² in the homogenized equation is zero.

Image of line L₁ in mirror line L₂. Method: reflect any two points on L₁ using the point-reflection formula, then find the line through their images.