Distance Formula
293. The Collinear Condition · when distances add up perfectly
AB + BC always equals AC when B lies on the segment AC.
Three points are collinear when they lie on one straight line. If B lies on segment AC, the parts add up: AB + BC = AC. (If B is off the line, AB + BC is greater than AC.)
What this lesson covers
Try to break it
Drag B along the line from A to C. AB + BC always equals AC — that's collinearity: no detour. Now imagine pulling B off the line: the path A→B→C becomes a detour and AB + BC strictly exceeds AC. Collinearity is the equality case of the triangle inequality.
How you build it
Place B on segment AC and show that AB + BC = AC.
- Click on the guide line near its left end to mark point A.
- Click on the guide line near its right end to mark point C.
- Click on the line between A and C to mark point B. B now lies on segment AC.
- Draw the segment from A to B. This is the first part, AB.
- Draw the segment from B to C. The two parts AB and BC together exactly cover AC — so AB + BC = AC.
The proof, step by step
Prove that A, B, and C are collinear when AB + BC = AC.
- Calculate AB using the distance formula: √[(x₂-x₁)² + (y₂-y₁)²]
- Calculate BC using the distance formula.
- Calculate AC using the distance formula.
- Verify that AB + BC = AC. If true, points are collinear.
Worked example
Points A(1, 1), B(3, 4), and C(5, 7) are collinear because:
Using the distance formula, AB = √[(3-1)²+(4-1)²] = √13, BC = √[(5-3)²+(7-3)²] = √13, AC = √[(5-1)²+(7-1)²] = √52 = 2√13. Since AB + BC = √13 + √13 = 2√13 = AC, the points are collinear.
- AB + BC = AC — correct
- AB + AC = BC
- AC + BC = AB
- AB = BC = AC