Find every triangle median accurately and quickly. Use sides or coordinates with complete input validation. Review centroid ratios, diagrams, exports, and saved history instantly.
Enter three sides or three coordinate points.
Load a tested example into the side-length form.
| Side a | Side b | Side c | Triangle type | Action |
|---|---|---|---|---|
| 3 | 4 | 5 | Right scalene | |
| 5 | 5 | 6 | Acute isosceles | |
| 8 | 8 | 8 | Equilateral | |
| 7 | 9 | 12 | Obtuse scalene | |
| 13 | 14 | 15 | Acute scalene |
A median joins one vertex to the opposite side midpoint. Every triangle always has exactly three medians.
These equations follow from Apollonius’s theorem. Each formula uses all three side lengths.
The area supports altitude, inradius, and circumradius calculations. Invalid triangles are rejected before these values appear.
A triangle median begins at a vertex. It ends at the opposite side midpoint. This construction divides the opposite side into equal segments.
All three medians meet at the centroid. The centroid is the triangle’s balance point. Uniform triangular material balances at this location.
The centroid divides every median predictably. The vertex section occupies two thirds. The midpoint section occupies one third.
A median targets the opposite midpoint. An altitude creates a perpendicular segment. An angle bisector divides one vertex angle equally.
These segments may overlap in special triangles. Equilateral triangles make all three constructions identical. Isosceles triangles create one shared symmetry segment.
The median formula subtracts the opposite side squared. A larger opposite side reduces the median radicand. Therefore, the longest side faces the shortest median.
This inverse relationship helps verify results. It also catches mislabeled side inputs. The summary identifies both extreme median lengths.
Coordinate mode first measures all three sides. It then calculates each opposite midpoint. Finally, it forms the three median line equations.
The coordinate centroid uses simple averaging. Its x-coordinate averages all vertex x-values. Its y-coordinate averages all vertex y-values.
Medians support geometric design and structural layout. They help locate balanced attachment points. They also appear in computer graphics and surveying.
Teachers use medians for theorem demonstrations. Students use them for triangle proofs. Designers use centroids when placing labels and shapes.
A median is a segment joining a vertex to the midpoint of the opposite side.
Every nondegenerate triangle has exactly three medians, one from each vertex.
Yes. All three medians meet at one common point called the centroid.
The centroid creates a 2:1 ratio. The vertex-to-centroid part is twice the remaining part.
Yes. This happens along symmetry lines in isosceles and equilateral triangles.
Yes. Equal sides create equal medians, altitudes, and angle bisectors.
Yes. Inputs such as 7/8, 3/2, and negative coordinate fractions are supported.
Three lengths must satisfy every triangle inequality. Otherwise, no closed triangle exists.
Yes. It finds each opposite midpoint and produces every median line equation.
When a suitable integer radicand exists, the calculator simplifies the square root symbolically.
Yes. CSV, JSON, PDF, printable, copied, shared, and SVG diagram exports are included.
The server stores up to 50 entries in this session.
No calculations are stored yet.
Server validation protects every submitted value. CSRF tokens protect normal browser submissions. Output escaping reduces injection risks.
The calculator uses double-precision floating-point arithmetic. PHP BCMath detection can support future extensions. Current geometric square roots still require floating-point evaluation.
The same file exposes optional JSON output. Submit the form to this page with ?api=1. The response contains success, errors, and raw results.
Built-in formula tests are available through ?self_test=1. They verify standard, equilateral, invalid, and coordinate cases.
JavaScript improves diagrams and exports. Core calculations still work without JavaScript. The server remains the final validation authority.
Generated by Advanced Triangle Median Calculator version 1.0.0.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.