The Central Idea
A coordinate proof translates geometric claims into algebraic statements that can be verified exactly.
The general pattern is:
\[ \boxed{ \text{geometric claim} \longrightarrow \text{coordinate representation} \longrightarrow \text{algebraic verification} \longrightarrow \text{geometric conclusion} } \]Coordinate Proof Is Still Geometry
The coordinates are a method, not the final subject.
A correct coordinate proof must connect each calculation back to a geometric property such as:
- parallel lines;
- perpendicular lines;
- equal lengths;
- midpoints;
- congruence;
- similarity;
- symmetry;
- polygon classification.
Why Coordinates Help
A diagram may suggest a relationship but cannot establish it by appearance alone.
Coordinates allow the relationship to be tested numerically and symbolically.
For example, two lines that look parallel can be proved parallel by showing that their slopes are equal.
Choosing a Coordinate System
A good coordinate proof begins with a useful placement.
You often have freedom to choose coordinates that simplify the algebra while preserving the geometric structure.
Strategic Placement
Common helpful choices include:
- placing one vertex at the origin;
- placing one side on the \(x\)-axis;
- centering a symmetric figure at the origin;
- using coordinates with simple parameters rather than arbitrary decimals.
Do Not Overspecialize the Figure
Coordinate choices should simplify the problem without adding properties that were not given.
If a quadrilateral is arbitrary, placing it as a rectangle would introduce right angles and parallel sides that may not belong to the original problem.
Use Parameters When Needed
A general figure often needs variable coordinates.
For example, a general point on the \(x\)-axis can be written
\[ (a,0). \]A rectangle aligned with the axes may be represented by
\[ (0,0),\quad (a,0),\quad (a,b),\quad (0,b). \]The Slope Formula
For distinct points
\[ A(x_1,y_1),\qquad B(x_2,y_2), \]with \(x_2\ne x_1\), the slope of \(\overleftrightarrow{AB}\) is
\[ \boxed{ m=\frac{y_2-y_1}{x_2-x_1}. } \]Slope Measures Direction
Slope records vertical change relative to horizontal change:
\[ m=\frac{\Delta y}{\Delta x}. \]Equal slopes indicate equal line direction for distinct nonvertical lines.
Parallel Lines by Slope
Two distinct nonvertical lines are parallel when they have equal slopes.
\[ \boxed{m_1=m_2\Rightarrow \ell_1\parallel\ell_2} \]provided the lines are distinct.
Vertical Parallel Lines
Vertical lines have undefined slope.
Two distinct vertical lines are parallel because both have equations of the form
\[ x=c. \]This case should be handled directly rather than forcing the slope formula.
Perpendicular Lines by Slope
For nonvertical, nonhorizontal perpendicular lines with slopes \(m_1\) and \(m_2\),
\[ \boxed{m_1m_2=-1}. \]Their slopes are negative reciprocals.
Horizontal and Vertical Perpendicular Lines
A horizontal line has slope
\[ 0. \]A vertical line has undefined slope.
Horizontal and vertical lines are perpendicular.
Example: Proving Parallelism
Let
\[ A(0,1),\quad B(4,3),\quad C(2,5),\quad D(6,7). \]Then
\[ m_{AB}=\frac{3-1}{4-0}=\frac12, \]and
\[ m_{CD}=\frac{7-5}{6-2}=\frac12. \]Therefore,
\[ \boxed{\overline{AB}\parallel\overline{CD}}. \]Example: Proving Perpendicularity
Suppose
\[ m_1=2 \]and
\[ m_2=-\frac12. \]Then
\[ m_1m_2=2\left(-\frac12\right)=-1. \]Therefore the lines are perpendicular.
The Distance Formula
For points
\[ A(x_1,y_1),\qquad B(x_2,y_2), \]the distance between them is
\[ \boxed{ AB= \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}. } \]Distance Comes from the Pythagorean Relationship
The horizontal and vertical changes form the legs of a right triangle:
\[ \Delta x=x_2-x_1, \] \[ \Delta y=y_2-y_1. \]Therefore,
\[ AB^2=(\Delta x)^2+(\Delta y)^2. \]Use Squared Distances When Possible
To prove two lengths equal, it is often enough to show their squared lengths are equal.
For nonnegative distances,
\[ AB^2=CD^2 \Rightarrow AB=CD. \]This avoids unnecessary square roots.
Example: Proving Equal Sides
Let
\[ A(0,0),\quad B(3,4),\quad C(6,0). \]Then
\[ AB^2=3^2+4^2=25 \]and
\[ BC^2=(6-3)^2+(0-4)^2=9+16=25. \]Therefore,
\[ \boxed{AB=BC}. \]Thus \(\triangle ABC\) is isosceles.
The Midpoint Formula
The midpoint of
\[ A(x_1,y_1) \]and
\[ B(x_2,y_2) \]is
\[ \boxed{ M\left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right). } \]Midpoints Prove Bisection
If two segments have the same midpoint, they bisect each other.
This is especially useful for proving that a quadrilateral is a parallelogram.
Example: Diagonals Bisect Each Other
Suppose quadrilateral \(ABCD\) has diagonals \(\overline{AC}\) and \(\overline{BD}\).
If
\[ M_{AC}=M_{BD}, \]then the diagonals bisect each other.
Therefore \(ABCD\) is a parallelogram.
Line Equations
A line may be represented by an equation such as
\[ y=mx+b \]or
\[ Ax+By=C. \]Line equations can encode slope, intercepts, incidence, and intersection conditions.
Point-Slope Form
A line through \((x_1,y_1)\) with slope \(m\) can be written
\[ \boxed{ y-y_1=m(x-x_1). } \]Proving Collinearity
Points are collinear if they lie on the same line.
Coordinate methods include:
- showing equal slopes between successive point pairs;
- showing every point satisfies the same line equation;
- using a determinant or area test.
Slope Test for Collinearity
For points \(A\), \(B\), and \(C\), if the relevant slopes are defined and
\[ m_{AB}=m_{BC}, \]then the three points are collinear.
Vertical Collinearity
If several points share the same \(x\)-coordinate, they lie on the same vertical line:
\[ x=c. \]This gives collinearity without using slope.
Area Test for Collinearity
Three points are collinear when the triangle they form has area
\[ 0. \]Coordinate formulas for signed area or determinants can test this condition.
Vector Difference between Points
From
\[ A(x_1,y_1) \]to
\[ B(x_2,y_2), \]the displacement vector is
\[ \boxed{ \overrightarrow{AB} = \langle x_2-x_1,\ y_2-y_1\rangle. } \]Vectors and Parallelism
Two nonzero vectors are parallel when one is a scalar multiple of the other.
If
\[ \overrightarrow{CD}=k\overrightarrow{AB}, \]then the corresponding directed segments have parallel directions.
Vectors and Perpendicularity
Vectors
\[ \mathbf{u}=\langle a,b\rangle \]and
\[ \mathbf{v}=\langle c,d\rangle \]are perpendicular when their dot product is zero:
\[ \boxed{ac+bd=0}. \]Why the Dot Product Helps
The dot-product test handles horizontal, vertical, and oblique directions uniformly.
It avoids special slope cases involving undefined slopes.
Example: Dot-Product Perpendicularity
Let
\[ \mathbf{u}=\langle3,4\rangle \]and
\[ \mathbf{v}=\langle4,-3\rangle. \]Then
\[ \mathbf{u}\cdot\mathbf{v} = 3(4)+4(-3) = 12-12 = 0. \]Therefore the vectors are perpendicular.
Proving a Quadrilateral Is a Parallelogram
Coordinate approaches include proving:
- both pairs of opposite sides are parallel;
- both pairs of opposite sides are congruent;
- one pair of opposite sides is both parallel and congruent;
- the diagonals bisect each other.
Parallelogram by Slopes
For quadrilateral \(ABCD\), show
\[ m_{AB}=m_{CD} \]and
\[ m_{BC}=m_{AD}. \]Then both pairs of opposite sides are parallel.
Parallelogram by Midpoints
Compute the midpoint of \(\overline{AC}\) and the midpoint of \(\overline{BD}\).
If they are equal, the diagonals bisect each other, so the quadrilateral is a parallelogram.
Proving a Rectangle
One efficient route is:
- prove the quadrilateral is a parallelogram;
- prove one angle is right.
Coordinate methods can establish the right angle by slopes or dot products.
Rectangle by Equal Diagonals
Another theorem says that a parallelogram with congruent diagonals is a rectangle.
Thus a coordinate proof can:
- prove the diagonals bisect each other;
- prove the diagonal lengths are equal.
Proving a Rhombus
Coordinate approaches include:
- showing all four side lengths are equal;
- showing the figure is a parallelogram with perpendicular diagonals;
- showing the figure is a parallelogram and one diagonal bisects an angle.
Rhombus by Equal Side Lengths
If
\[ AB=BC=CD=DA, \]then \(ABCD\) is a rhombus by definition.
Squared distances often make this computation efficient.
Proving a Square
A square is both a rectangle and a rhombus.
A coordinate proof may show:
- four equal sides and one right angle;
- a parallelogram with equal sides and equal diagonals;
- a rectangle with adjacent equal sides.
Do Not Prove More Than Necessary
To prove a square, you do not always need to calculate every side, every angle, both diagonals, and both midpoint relationships.
Choose a sufficient characterization and prove only what it requires.
Proving an Isosceles Trapezoid
One coordinate strategy is:
- show exactly one pair of opposite sides is parallel under the adopted trapezoid convention;
- show the two legs are congruent.
Equivalent characterizations may also be used when their hypotheses are met.
Trapezoid Convention Matters
Some definitions require a trapezoid to have exactly one pair of parallel sides. Others require at least one pair.
A coordinate proof should match the convention being used.
Proving a Kite
A kite can be established by showing two distinct pairs of adjacent sides are congruent.
Coordinate distance calculations provide a direct test.
Triangle Congruence by Coordinates
Coordinate geometry can provide the relationships needed for:
- SSS;
- SAS;
- ASA;
- AAS;
- HL.
The final congruence argument still uses the ordinary geometric criterion.
Coordinate SSS Example
If the three squared side lengths of one triangle match the three corresponding squared side lengths of another, then the side lengths themselves match.
Therefore SSS establishes congruence.
Triangle Similarity by Coordinates
Coordinate methods can establish:
- equal angles through slope relationships;
- proportional side lengths through distance formulas;
- a dilation relationship through coordinate scaling.
Similarity through Parallel Lines
If coordinate calculations prove that two lines are parallel, corresponding or alternate interior angle relationships may then establish AA similarity.
Coordinate and synthetic reasoning can therefore work together.
Coordinate Dilations
A dilation centered at the origin with scale factor \(k\) maps
\[ (x,y) \]to
\[ \boxed{(kx,ky)}. \]This gives a direct algebraic model of similarity.
Coordinate Translations
Translation by vector
\[ \langle a,b\rangle \]maps
\[ (x,y) \]to
\[ \boxed{(x+a,y+b)}. \]Coordinate Reflections
Common reflection rules include:
\[ (x,y)\mapsto(x,-y) \]across the \(x\)-axis,
\[ (x,y)\mapsto(-x,y) \]across the \(y\)-axis, and
\[ (x,y)\mapsto(y,x) \]across the line \(y=x\).
Coordinate Rotations about the Origin
Common exact rotation rules include:
\[ 90^\circ:\quad(x,y)\mapsto(-y,x), \] \[ 180^\circ:\quad(x,y)\mapsto(-x,-y), \] \[ 270^\circ:\quad(x,y)\mapsto(y,-x). \]Transformations Can Support Coordinate Proof
If one set of coordinates is mapped to another by a rigid transformation, the figures are congruent.
If a dilation is also required, the figures are similar.
Coordinate Proof of Symmetry
A figure has symmetry when a transformation maps the figure onto itself.
Coordinate rules allow every vertex to be checked explicitly.
Example: Reflection Symmetry
Suppose the vertices of a figure occur in pairs
\[ (a,b) \]and
\[ (-a,b). \]Reflection across the \(y\)-axis exchanges these points.
This supports symmetry about the \(y\)-axis.
Coordinate Proof of a Perpendicular Bisector
To prove line \(\ell\) is the perpendicular bisector of \(\overline{AB}\):
- find the midpoint of \(\overline{AB}\);
- show that midpoint lies on \(\ell\);
- show \(\ell\) is perpendicular to \(\overline{AB}\).
Locus Method for the Same Proof
Another route is to show that every point \(P(x,y)\) on \(\ell\) satisfies
\[ PA=PB. \]Then the perpendicular-bisector locus theorem identifies \(\ell\).
Coordinate Proof of an Angle Bisector
One method is to show that points on the candidate line are equidistant from the two sides of the angle.
Equal perpendicular distances characterize an angle bisector.
Distance from a Point to a Line
For line
\[ Ax+By+C=0, \]the distance from point \((x_0,y_0)\) is
\[ \boxed{ \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}. } \]Coordinate Proof of Tangency
A line tangent to a circle is perpendicular to the radius at the point of tangency.
Coordinate proof may therefore:
- verify the point lies on the circle;
- find the radius direction;
- show the line is perpendicular to that radius.
Example: Tangency to a Circle
For
\[ x^2+y^2=25 \]at point
\[ T(3,4), \]the radius slope is
\[ \frac43. \]A line through \(T\) with slope
\[ -\frac34 \]is perpendicular to the radius and therefore tangent.
Coordinate Proof of a Circle Center
If several points are claimed to lie on one circle, a candidate center \(O\) can be verified by showing equal distances:
\[ OA=OB=OC=\cdots. \]Circle Equation
A circle centered at \((h,k)\) with radius \(r\) has equation
\[ \boxed{ (x-h)^2+(y-k)^2=r^2. } \]Substitution as Incidence Proof
To prove a point lies on a line or circle, substitute its coordinates into the equation.
If the equation is satisfied, the point lies on the object.
Intersections by Solving Systems
Geometric intersections become algebraic systems.
For example, the intersection of two lines is found by solving their two equations simultaneously.
Intersection Count Matters
Algebra can reveal whether two geometric objects meet in:
- no points;
- one point;
- two points;
- infinitely many points.
These counts often have geometric meaning.
Coordinate Proof and Determinants
Determinants can encode several geometric tests.
They can help with:
- collinearity;
- signed area;
- orientation;
- parallel vector tests.
Signed Area of a Triangle
For points
\[ A(x_1,y_1),\quad B(x_2,y_2),\quad C(x_3,y_3), \]the signed double area can be written
\[ x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2). \]If this expression is zero, the points are collinear.
Area by Coordinates
The triangle area is
\[ \boxed{ \frac12 \left| x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2) \right|. } \]Coordinate Area as a Proof Tool
Equal-area claims, collinearity claims, and decomposition arguments can all be verified through coordinate area formulas.
Choosing between Slope and Vector Methods
Slopes are intuitive for many two-dimensional line problems.
Vectors are often more robust because they handle vertical directions without special undefined-slope cases.
Choosing between Distance and Squared Distance
If the proof only needs to compare lengths, squared distance is often cleaner.
Use the full distance only when an actual length value is needed.
Choosing between Midpoints and Slopes
For a parallelogram, checking diagonal midpoints may require less algebra than calculating four side slopes.
The shortest valid proof is usually preferable.
Coordinate Proof Should Match the Goal
Different goals call for different invariants.
- Parallelism: slope or scalar-multiple vectors.
- Perpendicularity: negative reciprocal slopes or zero dot product.
- Congruent segments: distance.
- Bisection: midpoint.
- Incidence: equation substitution.
- Symmetry: transformation rules.
Common Error: Using the Picture Instead of the Coordinates
A coordinate graph may be distorted by unequal axis scales.
A right angle may not look right and a circle may not look circular.
Trust the coordinates and equations.
Common Error: Dividing by Zero in the Slope Formula
Vertical lines have
\[ x_2-x_1=0. \]Their slope is undefined.
Handle them directly instead of dividing by zero.
Common Error: Saying Equal Slopes Prove Coincidence
Equal slopes show that two nonvertical lines have the same direction.
The lines may be parallel or may be the same line.
If distinctness matters, verify it separately.
Common Error: Misusing the Negative Reciprocal Rule
The product
\[ m_1m_2=-1 \]applies when both slopes are finite and nonzero.
Horizontal-vertical perpendicularity must be treated separately.
Common Error: Comparing Unsquared and Squared Lengths
If
\[ AB^2=25, \]then
\[ AB=5, \]not \(25\).
Keep squared distances and actual lengths distinct.
Common Error: Wrong Midpoint Coordinates
The midpoint averages corresponding coordinates:
\[ \left( \frac{x_1+x_2}{2}, \frac{y_1+y_2}{2} \right). \]Do not mix an \(x\)-coordinate with a \(y\)-coordinate.
Common Error: Proving a Special Case Only
If the theorem concerns every parallelogram, choosing coordinates for a rectangle proves only a special case.
The coordinate placement must preserve the generality of the claim.
Common Error: Adding Extra Symmetry
Centering a general figure at the origin may accidentally impose symmetry if coordinates are chosen carelessly.
Coordinate convenience should not strengthen the hypotheses.
Common Error: Calculating without Concluding
A list of slopes and distances is not yet a proof.
State what each calculation establishes geometrically and how that establishes the goal.
Common Error: Using Too Many Calculations
Coordinate proofs can become unnecessarily long.
Choose a theorem first, then compute only the quantities required by that theorem.
Common Error: Decimal Approximation Too Early
Exact fractions and radicals preserve exact relationships.
Replacing them with rounded decimals too early can obscure equality, proportionality, or perpendicularity.
Common Error: Ignoring Units
Coordinate distances inherit whatever units the coordinate system represents.
Area values use squared units.
Keep dimensional meaning consistent.
A Reliable Coordinate-Proof Workflow
- State the geometric goal.
- Choose coordinates that simplify the figure without adding assumptions.
- Identify the theorem or characterization that would prove the goal.
- Decide which coordinate tools are needed: slope, distance, midpoint, vector, equation, or area.
- Carry out exact algebra.
- Translate each result back into geometry.
- State the final geometric conclusion explicitly.
A Reliable Quadrilateral Workflow
- Identify the target classification.
- Choose one sufficient characterization of that class.
- Compute only the slopes, distances, or midpoints required.
- Verify every condition in the characterization.
- Conclude the classification.
A Reliable Triangle Workflow
- Identify the desired triangle property.
- Use distances for side relationships.
- Use slopes or vectors for angle relationships.
- Use midpoint calculations for medians or bisection.
- Apply the relevant triangle theorem or criterion.
Example: Prove a Quadrilateral Is a Parallelogram
Let
\[ A(0,0),\quad B(5,1),\quad C(7,4),\quad D(2,3). \]Compute:
\[ m_{AB}=\frac15, \] \[ m_{CD}=\frac{4-3}{7-2}=\frac15, \] \[ m_{BC}=\frac{4-1}{7-5}=\frac32, \] \[ m_{AD}=\frac{3-0}{2-0}=\frac32. \]Both pairs of opposite sides are parallel.
Therefore,
\[ \boxed{ABCD\text{ is a parallelogram}.} \]Example: Prove a Quadrilateral Is a Rectangle
Suppose a coordinate proof has already established that \(ABCD\) is a parallelogram.
If
\[ m_{AB}=\frac12 \]and
\[ m_{BC}=-2, \]then
\[ m_{AB}m_{BC}=-1. \]So adjacent sides are perpendicular.
Therefore the parallelogram has a right angle and is a rectangle.
Example: Prove an Isosceles Triangle
Let
\[ A(-3,0),\quad B(0,4),\quad C(3,0). \]Then
\[ AB^2=3^2+4^2=25 \]and
\[ BC^2=3^2+(-4)^2=25. \]Hence
\[ AB=BC. \]Therefore,
\[ \boxed{\triangle ABC\text{ is isosceles}.} \]Example: Prove a Segment Is a Median
In \(\triangle ABC\), suppose
\[ B(-2,1),\qquad C(6,5). \]Their midpoint is
\[ M\left( \frac{-2+6}{2}, \frac{1+5}{2} \right) = (2,3). \]Therefore any segment from \(A\) to \(M(2,3)\) is a median from vertex \(A\).
Example: Prove a Line Is a Perpendicular Bisector
Let
\[ A(-4,2),\qquad B(4,2). \]Their midpoint is
\[ (0,2). \]Segment \(\overline{AB}\) is horizontal, so the vertical line
\[ x=0 \]passes through the midpoint and is perpendicular to the segment.
Thus \(x=0\) is the perpendicular bisector.
Example: Prove Two Diagonals Bisect Each Other
Let
\[ A(0,0),\quad B(6,0),\quad C(8,4),\quad D(2,4). \]The midpoint of \(\overline{AC}\) is
\[ (4,2). \]The midpoint of \(\overline{BD}\) is also
\[ (4,2). \]Therefore the diagonals bisect each other, so \(ABCD\) is a parallelogram.
Example: Coordinate Similarity
Suppose
\[ A(0,0),\quad B(2,0),\quad C(0,3) \]and
\[ D(0,0),\quad E(6,0),\quad F(0,9). \]The second triangle is obtained from the first by dilation with scale factor
\[ 3. \]Therefore,
\[ \boxed{\triangle ABC\sim\triangle DEF}. \]Practice 1: Slope
Find the slope of the line through
\[ A(1,2) \]and
\[ B(5,10). \]Practice 2: Parallel Lines
Two distinct nonvertical lines have slopes
\[ \frac34 \]and
\[ \frac34. \]What relationship holds?
Practice 3: Perpendicular Lines
One line has slope
\[ -3. \]What slope would a nonvertical perpendicular line have?
Practice 4: Distance
Find the distance between
\[ (0,0) \]and
\[ (6,8). \]Practice 5: Midpoint
Find the midpoint of
\[ A(-4,6) \]and
\[ B(8,2). \]Practice 6: Equal Lengths
If
\[ AB^2=49 \]and
\[ CD^2=49, \]what can be concluded about \(AB\) and \(CD\)?
Practice 7: Collinearity
Points \(A\), \(B\), and \(C\) lie on the same nonvertical line if
\[ m_{AB}=m_{BC}. \]What geometric conclusion follows?
Practice 8: Dot Product
Determine whether
\[ \langle2,5\rangle \]and
\[ \langle5,-2\rangle \]are perpendicular.
Practice 9: Diagonal Midpoints
In a quadrilateral, both diagonals have midpoint
\[ (3,4). \]What can be concluded?
Practice 10: Identify the Error
A student proves that two lines have equal slopes and concludes that they are distinct parallel lines.
What additional issue must be checked?
Practice Answers
Practice 1
The slope is
\[ m= \frac{10-2}{5-1} = \frac84 = \boxed{2}. \]Practice 2
Distinct nonvertical lines with equal slopes are parallel:
\[ \boxed{\ell_1\parallel\ell_2}. \]Practice 3
The perpendicular slope is the negative reciprocal:
\[ \boxed{\frac13}. \]Indeed,
\[ (-3)\left(\frac13\right)=-1. \]Practice 4
The distance is
\[ \sqrt{6^2+8^2} = \sqrt{36+64} = \sqrt{100} = \boxed{10}. \]Practice 5
The midpoint is
\[ \left( \frac{-4+8}{2}, \frac{6+2}{2} \right) = \boxed{(2,4)}. \]Practice 6
Distances are nonnegative, so equal squared lengths imply equal lengths:
\[ \boxed{AB=CD=7}. \]Practice 7
Equal slopes show the three points lie on the same line:
\[ \boxed{A,\ B,\ C\text{ are collinear}.} \]Practice 8
Compute the dot product:
\[ 2(5)+5(-2)=10-10=0. \]Therefore,
\[ \boxed{\text{the vectors are perpendicular}.} \]Practice 9
The diagonals bisect each other because they have the same midpoint.
Therefore,
\[ \boxed{\text{the quadrilateral is a parallelogram}.} \]Practice 10
Equal slopes show equal direction, but the two lines could be the same line.
To conclude they are distinct parallel lines, distinctness must also be established.
What to Remember
Coordinate proof converts geometry into exact algebraic relationships and then translates the results back into geometric conclusions.
- Slope tests parallelism and, with care, perpendicularity.
- Vectors provide robust tests for direction and perpendicularity.
- Distance proves equal lengths and supports congruence.
- Midpoints prove bisection and are especially useful with diagonals.
- Line and circle equations test incidence and intersections.
- Coordinate transformations can establish congruence, similarity, and symmetry.
- Good coordinate choices simplify algebra without adding extra geometric assumptions.
- Exact fractions, radicals, and squared distances often keep proofs cleaner than decimals.
- Every calculation must be connected to the geometric property it establishes.
The essential habit is: choose the geometric theorem first, then compute only the coordinate quantities needed to verify its hypotheses.
Where We Go Next
Coordinate proof translates geometry into algebra. Another powerful approach proves relationships by tracking how figures behave under transformations.
The next lesson, Transformation-Based Proof, develops proofs using translations, rotations, reflections, dilations, invariants, and composition of transformations.