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Geometry, coordinates, surveying and trigonometry
Side split by an internal angle bisector
Distance from B to the point D where the bisector of the angle at A meets the opposite side BC.
- Ref
- GEO-14
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- BD [cm]
The angle bisector theorem says that the bisector of the angle at A meets the opposite side BC at a point D that splits it in the ratio of the two adjacent sides, BD : DC = c : b. With BC = a, the part next to B is BD = ac/(b + c) and DC = ab/(b + c). The same idea locates the incentre.
Distance from B to the point D where the bisector of the angle at A meets the opposite side BC. The bisector divides BC in the ratio c : b.
Calculator
- The other part, DC = ab/(b + c)2.8 cm
- Ratio BD : DC (equals c : b)1.5
- Check, BD + DC7 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The bisector from A meets BC 4.2 cm from B and 2.8 cm from C, in the ratio 6 : 4 of the two sides at A.
- If the two sides at A are equal the bisector hits the midpoint, which is the isosceles case.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm) | Local step | Change in result |
|---|---|---|---|---|---|---|
| a Side BC | 2.5 to 9.9 cm | 1.5 | 5.94 | 4.44 | +10% | +10 % |
| b Side CA | 1.5 to 12.5 cm | 2.27 | 5.6 | 3.33 | +10% | −3.85 % |
| c Side AB | 3.5 to 10.5 cm | 3.267 | 5.069 | 1.802 | +10% | +3.77 % |
Sweep of Side AB
| Side AB (cm) | Length BD (cm) |
|---|---|
| 3.5 | 3.267 |
| 4.375 | 3.657 |
| 5.25 | 3.973 |
| 6.125 | 4.235 |
| 7 | 4.455 |
| 7.875 | 4.642 |
| 8.75 | 4.804 |
| 9.625 | 4.945 |
| 10.5 | 5.069 |
Sweep one variable
Variables and how each one contributes
Length BD
What the equation gives you. Everything else feeds it.
Side BC
The length of the side BC that the bisector meets, in cm. A longer side lengthens BD in direct proportion. Raising it by 10 % (to 7.7) moves the result by +10 %, from 4.2 to 4.62 cm.
Side CA
The length of side CA, in cm. A longer side CA moves D towards B, so BD falls. Raising it by 10 % (to 4.4) moves the result by −3.85 %, from 4.2 to 4.038 cm.
Side AB
The length of side AB, in cm. A longer side AB moves D towards C, so BD rises. Raising it by 10 % (to 6.6) moves the result by +3.77 %, from 4.2 to 4.358 cm.
Worked example with the values above
- The ratio c : b = 6 : 4, so the share of BC next to B is c/(b + c) = 0.6.
- Multiply by a = 7 cm.
- BD = 4.2 cm.
Assumptions and limits
- A non-degenerate triangle: the three sides satisfy the strict triangle inequality.
- All lengths and the area use one consistent unit.
Common mistakes
- Swapping b and c: the part next to B is ac/(b + c), with c being the side AB.
- Using the opposite side in the ratio instead of the two sides at the angle being bisected.
- Confusing an internal bisector with an external one.
Statistics and uncertainty
The split depends only on the ratio of the two sides at the corner, so equal proportional errors in b and c cancel.
Where it is used
- Fitting the largest circle in a triangle and the circle through three points.
- Centre lines, balance points and bracing positions of triangular frames and plates.
Maths you need
- Triangle inequality: any two sides add to more than the third
- Heron’s formula: area from three sides
- Midpoint: the point halfway along a segment