Mathwords: Concurrent

Key Formula

\begin{vmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{vmatrix} = 0

Where:

$a_i, b_i, c_i$ = Coefficients of the i-th line written in the form a_i x + b_i y + c_i = 0
$i = 1, 2, 3$ = Index for each of the three lines being tested for concurrency

Worked Example

Problem: Determine whether the three lines x + y − 4 = 0, 2x − y − 2 = 0, and 3x + 2y − 10 = 0 are concurrent, and if so, find the point of concurrency.

Step 1: Write each line in the standard form a_i x + b_i y + c_i = 0 and identify the coefficients.

L_1: a_1=1,\; b_1=1,\; c_1=-4 \qquad L_2: a_2=2,\; b_2=-1,\; c_2=-2 \qquad L_3: a_3=3,\; b_3=2,\; c_3=-10

Step 2: Compute the 3×3 determinant. If it equals 0, the lines are concurrent.

\begin{vmatrix} 1 & 1 & -4 \\ 2 & -1 & -2 \\ 3 & 2 & -10 \end{vmatrix} = 1\bigl((-1)(-10)-(-2)(2)\bigr) - 1\bigl((2)(-10)-(-2)(3)\bigr) + (-4)\bigl((2)(2)-(-1)(3)\bigr)

Step 3: Evaluate each minor and simplify.

= 1(10+4) - 1(-20+6) + (-4)(4+3) = 14 - (-14) + (-28) = 14 + 14 - 28 = 0

Step 4: Since the determinant equals 0, the lines are concurrent. Find the point by solving any two of the equations. Solve L₁ and L₂ simultaneously.

x + y = 4 \quad \text{and} \quad 2x - y = 2 \;\Longrightarrow\; 3x = 6 \;\Longrightarrow\; x = 2,\; y = 2

Step 5: Verify (2, 2) satisfies L₃: 3(2) + 2(2) − 10 = 6 + 4 − 10 = 0. ✓

3(2) + 2(2) - 10 = 0 \;\checkmark

Answer: The three lines are concurrent, meeting at the point (2, 2).

Another Example

This example applies concurrency in a geometric setting (triangle medians) rather than testing three arbitrary lines with the determinant condition.

Problem: Show that the medians of a triangle with vertices A(0, 0), B(6, 0), and C(0, 6) are concurrent, and find the centroid.

Step 1: Find the midpoints of each side. Midpoint of BC is M_A, midpoint of AC is M_B, and midpoint of AB is M_C.

M_A = \left(\frac{6+0}{2},\frac{0+6}{2}\right) = (3,3), \quad M_B = \left(\frac{0+0}{2},\frac{0+6}{2}\right) = (0,3), \quad M_C = \left(\frac{0+6}{2},\frac{0+0}{2}\right) = (3,0)

Step 2: Write the equation of each median. Median from A(0,0) to M_A(3,3): y = x. Median from B(6,0) to M_B(0,3): slope = (3−0)/(0−6) = −1/2, so y − 0 = −½(x − 6).

\text{Median } AM_A:\; y = x \qquad \text{Median } BM_B:\; y = -\tfrac{1}{2}x + 3

Step 3: Solve the first two medians simultaneously to find their intersection.

x = -\tfrac{1}{2}x + 3 \;\Longrightarrow\; \tfrac{3}{2}x = 3 \;\Longrightarrow\; x = 2,\; y = 2

Step 4: Write the third median from C(0,6) to M_C(3,0): slope = (0−6)/(3−0) = −2, so y − 6 = −2(x − 0), giving y = −2x + 6. Check (2, 2): −2(2) + 6 = 2. ✓

y = -2(2) + 6 = 2 \;\checkmark

Answer: The three medians are concurrent at the centroid (2, 2), which equals ((0+6+0)/3, (0+0+6)/3).

Frequently Asked Questions

What is the difference between concurrent lines and intersecting lines?

Any two non-parallel lines in a plane intersect at a point, so 'intersecting' usually refers to two lines meeting. 'Concurrent' specifically means three or more lines all share the same single point. Two lines alone are not typically called concurrent.

What are common examples of concurrent lines in a triangle?

Every triangle has four famous sets of concurrent lines. The three medians meet at the centroid, the three altitudes meet at the orthocenter, the three perpendicular bisectors meet at the circumcenter, and the three angle bisectors meet at the incenter. Each of these is a point of concurrency.

How do you prove three lines are concurrent?

The most common algebraic method is to compute the 3×3 determinant of the coefficients when each line is written as ax + by + c = 0. If the determinant equals zero, the lines are concurrent. Alternatively, you can solve two of the equations to find their intersection and then verify that the third line passes through that same point.

Concurrent vs. Coincident

	Concurrent	Coincident
Definition	Three or more lines passing through a single common point	Two or more lines that lie exactly on top of each other (same line)
Number of shared points	Exactly one point	Infinitely many points (every point on the line)
Lines involved	Distinct lines that happen to share one point	Lines that are identical — same slope and same intercept
Test (for three lines)	Determinant of coefficient matrix = 0, but lines are not identical	All three equations reduce to the same equation

Why It Matters

Concurrency appears throughout geometry courses, especially when studying triangles: the centroid, incenter, circumcenter, and orthocenter are all defined as points of concurrency. In coordinate geometry and analytic proofs, the determinant test for concurrency is a standard technique on exams. Understanding concurrency also lays the groundwork for topics like Ceva's theorem and projective geometry.

Common Mistakes

Mistake: Calling two intersecting lines 'concurrent.'

Correction: Concurrency requires three or more lines (or curves) meeting at a single point. Two lines simply 'intersect' — the term concurrent is reserved for three or more.

Mistake: Assuming that if each pair of three lines intersects, the three lines must be concurrent.

Correction: Three lines can intersect in pairs at three different points, forming a triangle. They are concurrent only when all three pairwise intersections coincide at one point. Always verify with the determinant test or by checking the third line through the intersection of the first two.

Related Terms

Line — The basic object tested for concurrency
Curve — Curves can also be concurrent at a point
Point — The shared location where concurrent lines meet
Coincident — Lines that overlap entirely, distinct from concurrent
Centroid — Point of concurrency of a triangle's medians
Circumcenter — Point of concurrency of perpendicular bisectors
Incenter — Point of concurrency of angle bisectors