13.3 Proofs using coordinate geometry

Coordinate geometry proofs blend algebra and geometry, letting you prove geometric theorems using coordinates and equations. Instead of relying on purely visual reasoning, you place figures on a coordinate plane and use formulas for slope, distance, and midpoints to establish results rigorously.

This section covers proving properties of lines and segments, triangle centers, triangle congruence and similarity, and quadrilateral classification.

Coordinate Geometry Proofs

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Geometric theorems in coordinate geometry

The general strategy for any coordinate proof is:

1. Place the figure on a coordinate plane (choose coordinates that simplify your algebra).
2. Use formulas (midpoint, distance, slope) to calculate the quantities you need.
3. Show algebraically that the geometric property holds.

Lines, segments, and angles

• Midpoint formula: $\displaystyle (x_m, y_m) = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ gives the midpoint of the segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$. You'll use this constantly to prove that a point bisects a segment.
• Perpendicular bisector theorem: A point lies on the perpendicular bisector of a segment if and only if it is equidistant from the segment's endpoints. To prove this with coordinates, compute the distance from the point to each endpoint using the distance formula and show the two distances are equal.
• Parallel lines have equal slopes. If you calculate the slopes of two lines and they match, the lines are parallel.
• Perpendicular lines have slopes that are negative reciprocals of each other. If one line has slope $m$, a line perpendicular to it has slope $-\frac{1}{m}$ (provided $m \neq 0$). For a horizontal and vertical line pair, one slope is 0 and the other is undefined.

Triangle centers

Coordinate geometry gives you a clean way to locate and verify triangle centers:

• Centroid: The intersection of the three medians (each median connects a vertex to the midpoint of the opposite side). The centroid divides every median in a 2:1 ratio, with the longer segment closer to the vertex. You can also compute it directly: if the triangle has vertices $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, the centroid is at $\displaystyle \left(\frac{x_1+x_2+x_3}{3},\;\frac{y_1+y_2+y_3}{3}\right)$.
• Orthocenter: The intersection of the three altitudes. An altitude runs from a vertex perpendicular to the opposite side. To find it with coordinates, write the equations of two altitudes (using the negative reciprocal of the opposite side's slope) and solve the system.
• Incenter: The intersection of the three angle bisectors. Each angle bisector divides its angle into two equal parts. The incenter is equidistant from all three sides of the triangle.

Congruence and similarity via slope-distance

To prove triangles congruent or similar on the coordinate plane, you rely on the distance formula and, where needed, slope.

$\displaystyle d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Congruence: Two triangles are congruent if all three pairs of corresponding sides have equal lengths (SSS). Compute the six side lengths with the distance formula and match them up.

Similarity: Two triangles are similar if all three pairs of corresponding sides are proportional (SSS similarity). Compute the side lengths and check that the ratios are equal. For example, if triangle $ABC$ has sides 3, 4, 5 and triangle $DEF$ has sides 6, 8, 10, every ratio equals 2, so the triangles are similar with scale factor 2.

Slope plays a supporting role here. If you need to show two segments are parallel (for AA similarity arguments, for instance), showing equal slopes does the job. But note: congruent or similar triangles do not need to have matching slopes for corresponding sides. Two congruent triangles can be rotated relative to each other. Slope helps you verify parallelism and perpendicularity, not congruence directly.

Quadrilateral properties through coordinates

Coordinate proofs are especially useful for classifying quadrilaterals. The strategy is to test progressively stronger conditions.

Parallelogram (prove any one of these):

• Opposite sides have equal slopes (both pairs parallel)
• Opposite sides have equal lengths (use the distance formula)
• Diagonals bisect each other (both diagonals share the same midpoint)

Rectangle (first show it's a parallelogram, then prove one more property):

• Adjacent sides are perpendicular (their slopes are negative reciprocals)
• Diagonals are equal in length

Note: diagonals of a rectangle are not necessarily perpendicular. That's a common mistake. Perpendicular diagonals belong to rhombi, not rectangles in general.

Rhombus (first show it's a parallelogram, then):

• All four sides are equal in length
• Equivalently, diagonals are perpendicular

Square (satisfies the conditions for both a rectangle and a rhombus):

• All four sides are equal in length and all angles are right angles
• Equivalently: diagonals are equal in length, bisect each other, and are perpendicular

When writing a coordinate proof for any quadrilateral, label your vertices in order, compute the relevant slopes, distances, and midpoints, and state clearly which property each calculation establishes.