Measure bisected segments with theorem relationships. Review checks, graph ratios, and export clean example outputs. Useful for homework, proofs, exams, tutoring, and self study.
The triangle angle bisector theorem states that when an angle in a triangle is bisected, the opposite side is divided into two segments proportional to the two adjacent sides. In triangle ABC, if AD bisects angle A and meets BC at D, then BD divided by DC equals AB divided by AC.
This calculator helps with several advanced geometry tasks. You can split the base into correct proportional parts, recover a missing adjacent side from segment data, compute the internal bisector length, or test whether measured values satisfy the theorem within a chosen tolerance.
The tool is useful for geometry homework, proof verification, exam practice, design sketches, and educational demonstrations. It also gives graph output for quick comparison of sides and segments. That visual check helps you spot ratio patterns immediately.
Because triangle work depends on consistent measurements, the calculator checks for positive values and validates triangle side combinations when needed. When values are realistic, the results show not only the main answer, but also ratio checks, segment totals, shares of the base, and other useful supporting information.
Core theorem: BD / DC = AB / AC
Segment formulas: BD = BC × AB / (AB + AC), and DC = BC × AC / (AB + AC)
Missing side formulas: AB = AC × BD / DC, and AC = AB × DC / BD
Bisector length: AD = √[AB × AC × ((AB + AC)2 − BC2)] / (AB + AC)
These formulas assume all lengths use the same unit and the side values form a valid triangle whenever full side geometry is required.
| Mode | Inputs | Output |
|---|---|---|
| Solve base segments | AB = 6, AC = 9, BC = 10 | BD = 4, DC = 6 |
| Solve missing side | AC = 12, BD = 5, DC = 8 | AB = 7.5 |
| Find bisector length | AB = 6, AC = 9, BC = 10 | AD ≈ 5.4772 |
| Check consistency | AB = 7.5, AC = 12, BD = 5, DC = 8 | Ratios match exactly |
It says an internal angle bisector divides the opposite side in the same ratio as the two adjacent sides.
Yes. Enter AB, AC, and BC. The calculator splits BC into BD and DC using the proportional theorem formula.
Yes. Enter the known adjacent side and the two base segments. The missing side is recovered from the matching ratio.
Some outputs, especially bisector length, require side values that can actually form a triangle. Invalid combinations can produce meaningless geometry.
Yes. Keep every length in one unit system. Mixing centimeters and meters will distort the theorem ratio and final results.
It compares AB divided by AC with BD divided by DC. Small differences show measurement error or an incorrect setup.
Yes. The calculator accepts decimals and lets you choose output precision for cleaner classroom or project reporting.
The graph visualizes calculated sides, segments, or ratio comparisons. It makes proportional relationships easier to inspect quickly.
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.