Triangle, centroid, circumcenter, orthocenter, and Euler lineLive current inputs
Math & Statistics
Triangle Graphing Calculator
Graph a triangle from three Cartesian vertices and calculate side lengths, signed and absolute area, orientation, centroid, circumcenter, and orthocenter from the same coordinates.
Coordinate area-
Vertex orientation-
Centroid G-
Circumcenter O-
Orthocenter H-
Side AB-
Side BC-
Side CA-
COORDINATE GEOMETRY BOARD
Triangle, centroid, circumcenter, orthocenter, and Euler line
A responsive Cartesian grid frames the entered vertices. Distinct markers identify G, O, and H; dashed altitudes and the Euler line reveal their geometric relationships.
COORDINATE CONSTRUCTION REGISTER
Vectors, lengths, slopes, and centers derived from the vertices
The table preserves both coordinate formulas and geometric meaning.
| Object | Coordinate or vector | Magnitude or slope | Construction rule | Diagnostic |
|---|
POINT ENTRY
Enter vertices in the intended traversal order
- Record x and y for each vertex.
- Use one coordinate system and datum.
- Preserve the desired A-B-C order.
- Avoid three collinear points.
- Use the plotted bounds to spot coordinate transposition.
CENTER RELATIONSHIPS
Different centers answer different geometric questions
The centroid balances the three vertices. The circumcenter is equidistant from them and may lie outside an obtuse triangle. The orthocenter is where the three altitudes meet.
For a non-equilateral triangle, O, G, and H lie on the Euler line, with G dividing OH in a 1:2 ratio.
ANALYTIC GEOMETRY
Use the cross product for area and perpendicular bisectors for the circumcenter
The signed shoelace cross product gives orientation and twice the area. The centroid is an average; the circumcenter solves equal-distance equations; the orthocenter follows H = A + B + C - 2O.
Detailed calculation process and general formulas
2K_signed = (Bx-Ax)(Cy-Ay) - (By-Ay)(Cx-Ax)G = (A + B + C) / 3|AB| = √((Bx-Ax)² + (By-Ay)²)O: |O-A|² = |O-B|² = |O-C|²H = A + B + C - 2OSymbols, meanings, and units
- A,B,C
- entered Cartesian verticescoordinate units
- K_signed
- oriented triangle areacoordinate units²
- G
- centroidcoordinate units
- O
- circumcentercoordinate units
- H
- orthocentercoordinate units
GRAPH DIAGNOSTICS
The plot exposes mistakes a value list can hide
Position and orientation matter.
Traversal
-Signed area distinguishes clockwise from counterclockwise vertex order.
Balance point
-The centroid is always inside a nondegenerate triangle.
Triangle type clue
-An outside circumcenter or orthocenter indicates obtuse geometry.
Decision takeaway: If the plot looks mirrored or stretched unexpectedly, verify coordinate order and units before using downstream values.
DATA-INTEGRITY CHECKS
Coordinate records that deserve a second look
- Swapped x and y
- Mixed coordinate reference systems
- Rounded survey points
- Nearly collinear vertices
- Axis units with unequal scale
- Clockwise order mistaken for an error
Applied decisions
How vertex placement moves the triangle centers
Acute triangle
All angles remain below 90°.
What the result clarifies: Centroid, circumcenter, and orthocenter all lie inside the triangle.
Obtuse triangle
One vertex moves past a right-angle boundary.
What the result clarifies: Circumcenter and orthocenter move outside while the centroid stays inside.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The graph uses Euclidean Cartesian coordinates with equal mathematical axis scale. It does not transform geographic latitude-longitude data or account for map projection, elevation, or surveying covariance.
Triangle Graphing Calculator | Coordinate Geometry, Centers, and Altitudes FAQ
Why is the orientation clockwise?
The signed cross product is negative for the entered A-B-C traversal.
Can a center lie outside the triangle?
Yes. Circumcenter and orthocenter lie outside an obtuse triangle.
What happens for collinear points?
Area is zero and unique circumcenter and orthocenter values are unavailable.
Why is the centroid always an average?
Triangle medians meet at the arithmetic mean of the three vertex coordinates.