Enter triangle side lengths to compute the angle bisector and segment ratios
Side AB (adjacent 1)
Side AC (adjacent 2)
Base Segment BD
Angle A (degrees)
Result
Side Ratio (AB : AC)
Base Segment DC
Bisector Length
Detailed Derivation
Angle Bisector Theorem Formula
BD / DC = AB / AC
Angle Bisector Length: l = (2bc*cos(A/2)) / (b + c)
l^2 = bc - mn (m, n = base segments)
The Angle Bisector Theorem is a fundamental result in triangle geometry. It states that the internal angle bisector divides the opposite side into segments proportional to the adjacent sides.
⚠The angle A is the vertex angle at the bisected vertex. Enter the angle in degrees (0-180). Results assume a valid triangle according to the triangle inequality.
What Is the Angle Bisector Theorem?
The Angle Bisector Theorem describes how an angle bisector in a triangle divides the opposite side. It is widely used in geometry proofs, construction problems, and real-world measurement applications.
Proportionality
BD/DC = AB/AC. The ratio of the base segments equals the ratio of the adjacent sides.
Bisector Length
The bisector length depends on adjacent sides and the vertex angle. It can be calculated using trigonometric or algebraic formulas.
Converse Theorem
If a line divides the opposite side proportionally to the adjacent sides, then it is an angle bisector. Useful for proving angle bisectors.
Exterior Version
The external angle bisector also follows a proportional rule, dividing the opposite side externally in the ratio of adjacent sides.
Teaching Example: Triangle ABC with AB=8, AC=12, BD=4, angle A=60. Ratio AB:AC = 8:12 = 2:3. DC = BD x AC/AB = 4 x 12/8 = 6. Bisector length l = (2x8x12xcos30)/(8+12) = 7.76.
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