Draw a triangle, then a line from each vertex to the midpoint of the opposite side. They will not form a messy three-way near-miss. They cross at exactly one point, every single time, no matter how lopsided you make the triangle. That guaranteed meeting is what geometry calls a point of concurrency.
When three or more lines pass through the same point, the lines are concurrent, and that shared point is the point of concurrency. In a triangle, four sets of special lines each meet this way, and each meeting point has its own name and its own job. Lines that share a point are concurrent; this is different from intersecting lines, where only two lines cross.
The Four Points of Concurrency
A triangle hides four of these meeting points. Each is built from a different set of three lines.
Centroid — built from the three medians (each median runs from a vertex to the midpoint of the opposite side). The centroid is the triangle's balance point, where a cardboard triangle would sit level on a pin.
Incenter — built from the three angle bisectors (each splits a vertex angle in half). The incenter is equidistant from the three sides and is the centre of the largest circle that fits inside the triangle.
Circumcenter — built from the three perpendicular bisectors of the sides. The circumcenter is equidistant from the three vertices and is the centre of the circle that passes through all of them.
Orthocenter — built from the three altitudes (each runs from a vertex perpendicular to the opposite side). The orthocenter has no single "centre" intuition; it is simply where the three heights meet.
You will meet each of these on its own — the circumcenter, the incenter, the centroid via the median of a triangle, and the orthocenter via the altitude of a triangle. Each is built from a sibling concept: the angle bisector builds the incenter, and the perpendicular bisector builds the circumcenter.
Does A Point of Concurrency Always Sit Inside The Triangle?
Not always. The centroid and incenter always sit inside any triangle. The circumcenter and orthocenter can sit inside, on, or outside the triangle depending on whether it is acute, right, or obtuse.
Examples of Point of Concurrency in a Triangle
These examples build from naming the point, to locating it on a grid, to the special triangle where all four collapse into one.
Example 1
Name the point of concurrency formed by the three medians of a triangle.
A median joins a vertex to the midpoint of the opposite side. The three medians meet at one point.
That point is the centroid.
Final answer: The centroid.
Example 2
A student is told the circumcenter is "the middle of the triangle," so they find the centroid instead. Where does this go wrong?
The intuitive move is to treat every triangle centre as "the middle," so the student averages the three vertices — that gives the centroid, the balance point.
Test it against the definition the question actually needs. The circumcenter must be equidistant from the three vertices so that one circle passes through all of them. The centroid is not equidistant from the vertices; it sits closer to the side it balances. So calling the centroid the circumcenter fails the equidistant check.
The rescue: match the point to its property, not to a vague "middle." Equidistant from the vertices means circumcenter (built from perpendicular bisectors). Equidistant from the sides means incenter (built from angle bisectors).
Final answer: The two points are different; only the circumcenter is equidistant from all three vertices.
Example 3
Find the centroid of the triangle with vertices A(0, 0), B(6, 0), and C(0, 9).
The centroid is the average of the three vertices' coordinates.
$$x = \frac{0 + 6 + 0}{3} = 2$$
$$y = \frac{0 + 0 + 9}{3} = 3$$
Final answer: The centroid is at $(2, 3)$.
Example 4
A triangle's incenter is equidistant from its three sides. If that distance is 4 cm, what circle is centred at the incenter?
The incenter is the centre of the circle that fits snugly inside the triangle, touching each side once. That circle is the incircle (inscribed circle).
Its radius equals the distance from the incenter to a side, which is 4 cm.
Final answer: The incircle, with radius 4 cm.
Example 5
For a right triangle, where is the circumcenter?
The circumcenter is equidistant from all three vertices. In a right triangle, that point lands exactly on the midpoint of the hypotenuse.
You can see why: the hypotenuse becomes a diameter of the circumcircle, and the right angle sits on the circle. So the centre is the midpoint of that diameter, which you can find with the midpoint formula.
Final answer: At the midpoint of the hypotenuse.
Example 6
In an equilateral triangle, how many distinct points of concurrency are there?
In a general triangle the centroid, incenter, circumcenter, and orthocenter are four separate points.
In an equilateral triangle, the three medians, angle bisectors, perpendicular bisectors, and altitudes are all the same three lines. So all four points land on top of each other.
Final answer: Just one — all four points of concurrency coincide.
Why A Triangle Guarantees These Meeting Points
"Three lines through one point — guaranteed."
That guarantee is the whole reason these points matter. In general, three random lines in a plane form a small triangle of near-misses, not a single crossing. The fact that the medians (or bisectors, or altitudes) of a triangle always land on one point is a genuine theorem, not a coincidence — and it is what lets engineers and designers rely on these centres.
The circumcenter solves a real placement problem. Suppose three towns sit at the corners of a triangle and need one shared water tower equidistant from all three. The circumcenter is the only point that distance can be equal to all three vertices — so it is exactly where the tower goes.
The centroid is a balance point. Cut a triangle from stiff card and it balances on a pin placed at the centroid. This is why the centroid is also called the centre of mass, and why it appears in structural engineering and physics.
The incenter fits the biggest inside circle. A round table that must fit inside a triangular alcove is centred at the incenter, with the incircle as its largest possible footprint.
The early study of these centres traces back to Euclid's Elements around 300 BCE, where the constructions for inscribed and circumscribed circles were first set out formally (Euclid's Elements, Book IV).
Mistakes That Mix Up The Four Centres
Mistake 1: Treating every triangle centre as "the middle"
Where it slips in: When a problem just says "find the centre of the triangle" and the reader picks whichever point comes to mind.
Don't do this: Assume there is one centre. The memorizer who learned "centroid = average of vertices" will reach for it on every problem, including ones that ask for the circumcenter.
The correct way: Read which property the problem needs. Equidistant from vertices means circumcenter; equidistant from sides means incenter; balance point means centroid; meeting of heights means orthocenter.
Mistake 2: Confusing the line that builds each point
Where it slips in: At the construction step, when the reader draws the wrong set of three lines.
Don't do this: Draw medians when the problem asked for the circumcenter. Medians go vertex-to-midpoint; perpendicular bisectors go through the midpoint at a right angle to the side. These are different lines and give different points.
The correct way: Pin the line to the point first. The first-instinct error here is reaching for the median because it is the easiest line to draw — pause and ask which line the named point actually requires.
Mistake 3: Assuming the point always sits inside the triangle
Where it slips in: On obtuse triangles, where the circumcenter and orthocenter leave the triangle entirely.
Don't do this: Restrict your search to the interior. The second-guesser who finds the circumcenter outside an obtuse triangle often erases it, assuming the answer must be wrong.
The correct way: Let the construction lead. For an obtuse triangle, the perpendicular bisectors genuinely meet outside the triangle, and that outside point is the correct circumcenter.
Conclusion
A point of concurrency in a triangle is the single point where three special lines meet.
The four classic points are the centroid (medians), incenter (angle bisectors), circumcenter (perpendicular bisectors), and orthocenter (altitudes).
The centroid is the balance point; the incenter is equidistant from the sides; the circumcenter is equidistant from the vertices.
The circumcenter and orthocenter can fall outside an obtuse triangle, while the centroid and incenter always stay inside.
In an equilateral triangle, all four points collapse into one.
To go further with a teacher, explore Bhanzu's geometry tutor sessions or a high school math tutor, and for younger learners the structured math classes online cover triangle constructions step by step.
Practice These To Solidify Your Understanding
Work through these, then check your method against the examples above.
Name the point of concurrency built from the three altitudes. (Answer to Question 1: the orthocenter.)
Find the centroid of the triangle with vertices $(0,0)$, $(3,0)$, $(0,6)$. (Answer to Question 2: $(1, 2)$.)
A right triangle has its right angle at vertex $C$. Where is its circumcenter? (Answer to Question 3: at the midpoint of the hypotenuse, the side opposite $C$.)
If you get stuck on Question 2, return to Example 3 and average the coordinates.
Want a live Bhanzu trainer to walk through triangle constructions with you? Book a free demo class.
Read More
Right angled triangle — where the circumcenter lands on the hypotenuse.
Triangle sum theorem — why a triangle's angles total 180°.
Properties of a triangle — the foundational facts behind these centres.
Types of triangles — how triangle type shifts where each centre sits.
Angle bisector theorem — the rule the incenter rests on.
Equation of a circle — used with the circumcircle and incircle.
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