Angle bisector theorem

Last updated
The theorem states for any triangle [?] DAB and [?] DAC where AD is a bisector, then
|
B
D
|
:
|
C
D
|
=
|
A
B
|
:
|
A
C
|
.
{\displaystyle |BD|:|CD|=|AB|:|AC|.} Triangle ABC with bisector AD.svg
The theorem states for any triangle DAB and DAC where AD is a bisector, then

In geometry, the angle bisector theorem is concerned with the relative lengths of the two segments that a triangle's side is divided into by a line that bisects the opposite angle. It equates their relative lengths to the relative lengths of the other two sides of the triangle.

Contents

Note that this theorem is not to be confused with the Inscribed Angle Theorem, which also involves angle bisection (but of an angle of a triangle inscribed in a circle).

Theorem

Consider a triangle ABC. Let the angle bisector of angle A intersect side BC at a point D between B and C. The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of the length of side AB to the length of side AC:

and conversely, if a point D on the side BC of ABC divides BC in the same ratio as the sides AB and AC, then AD is the angle bisector of angle A.

The generalized angle bisector theorem states that if D lies on the line BC, then

This reduces to the previous version if AD is the bisector of BAC. When D is external to the segment BC, directed line segments and directed angles must be used in the calculation.

The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof.

An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.

Proofs

There exist many different ways of proving the angle bisector theorem. A few of them are shown below.

Proof using similar triangles

Animated illustration of the angle bisector theorem. Animated illustration of angle bisector theorem.gif
Animated illustration of the angle bisector theorem.

As shown in the accompanying animation, the theorem can be proved using similar triangles. In the version illustrated here, the triangle gets reflected across a line that is perpendicular to the angle bisector , resulting in the triangle with bisector . The fact that the bisection-produced angles and are equal means that and are straight lines. This allows the construction of triangle that is similar to . Because the ratios between corresponding sides of similar triangles are all equal, it follows that . However, was constructed as a reflection of the line , and so those two lines are of equal length. Therefore, , yielding the result stated by the theorem.

Proof using Law of Sines

In the above diagram, use the law of sines on triangles ABD and ACD:

Angles ADB and ADC form a linear pair, that is, they are adjacent supplementary angles. Since supplementary angles have equal sines,

Angles DAB and DAC are equal. Therefore, the right hand sides of equations ( 1 ) and ( 2 ) are equal, so their left hand sides must also be equal.

which is the angle bisector theorem.

If angles DAB, ∠ DAC are unequal, equations ( 1 ) and ( 2 ) can be re-written as:

Angles ADB, ∠ ADC are still supplementary, so the right hand sides of these equations are still equal, so we obtain:

which rearranges to the "generalized" version of the theorem.

Proof using triangle altitudes

Bisekt.svg

Let D be a point on the line BC, not equal to B or C and such that AD is not an altitude of triangle ABC.

Let B1 be the base (foot) of the altitude in the triangle ABD through B and let C1 be the base of the altitude in the triangle ACD through C. Then, if D is strictly between B and C, one and only one of B1 or C1 lies inside ABC and it can be assumed without loss of generality that B1 does. This case is depicted in the adjacent diagram. If D lies outside of segment BC, then neither B1 nor C1 lies inside the triangle.

DB1B, ∠ DC1C are right angles, while the angles B1DB, ∠ C1DC are congruent if D lies on the segment BC (that is, between B and C) and they are identical in the other cases being considered, so the triangles DB1B, △DC1C are similar (AAA), which implies that:

If D is the foot of an altitude, then,

and the generalized form follows.

Proof using triangle areas

a
=
[?]
B
A
C
2
=
[?]
B
A
D
=
[?]
C
A
D
{\textstyle \alpha ={\frac {\angle BAC}{2}}=\angle BAD=\angle CAD} Angle bisector proof.svg

A quick proof can be obtained by looking at the ratio of the areas of the two triangles BAD, △CAD, which are created by the angle bisector in A. Computing those areas twice using different formulas, that is with base and altitude h and with sides a, b and their enclosed angle γ, will yield the desired result.

Let h denote the height of the triangles on base BC and be half of the angle in A. Then

and

yields

Length of the angle bisector

Diagram of Stewart's theorem Stewarts theorem.svg
Diagram of Stewart's theorem

The length of the angle bisector can be found by ,

where is the constant of proportionality of from the angle bisector theorem.

Proof: By Stewart's theorem, we have

Exterior angle bisectors

exterior angle bisectors (dotted red):
Points D, E, F are collinear and the following equations for ratios hold:

|
E
B
|
|
E
C
|
=
|
A
B
|
|
A
C
|
{\displaystyle {\tfrac {|EB|}{|EC|}}={\tfrac {|AB|}{|AC|}}}
,
|
F
B
|
|
F
A
|
=
|
C
B
|
|
C
A
|
{\displaystyle {\tfrac {|FB|}{|FA|}}={\tfrac {|CB|}{|CA|}}}
,
|
D
A
|
|
D
C
|
=
|
B
A
|
|
B
C
|
{\displaystyle {\tfrac {|DA|}{|DC|}}={\tfrac {|BA|}{|BC|}}} Aussenwinkelhalbierende2.svg
exterior angle bisectors (dotted red):
Points D, E, F are collinear and the following equations for ratios hold:
, ,

For the exterior angle bisectors in a non-equilateral triangle there exist similar equations for the ratios of the lengths of triangle sides. More precisely if the exterior angle bisector in A intersects the extended side BC in E, the exterior angle bisector in B intersects the extended side AC in D and the exterior angle bisector in C intersects the extended side AB in F, then the following equations hold: [1]

, ,

The three points of intersection between the exterior angle bisectors and the extended triangle sides D, E, F are collinear, that is they lie on a common line. [2]

History

The angle bisector theorem appears as Proposition 3 of Book VI in Euclid's Elements. According to Heath (1956 , p. 197 (vol. 2)), the corresponding statement for an external angle bisector was given by Robert Simson who noted that Pappus assumed this result without proof. Heath goes on to say that Augustus De Morgan proposed that the two statements should be combined as follows: [3]

If an angle of a triangle is bisected internally or externally by a straight line which cuts the opposite side or the opposite side produced, the segments of that side will have the same ratio as the other sides of the triangle; and, if a side of a triangle be divided internally or externally so that its segments have the same ratio as the other sides of the triangle, the straight line drawn from the point of section to the angular point which is opposite to the first mentioned side will bisect the interior or exterior angle at that angular point.

Applications

This theorem has been used to prove the following theorems/results:

Related Research Articles

<span class="mw-page-title-main">Quadrilateral</span> Polygon with four sides and four corners

In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words quadri, a variant of four, and latus, meaning "side". It is also called a tetragon, derived from Greek "tetra" meaning "four" and "gon" meaning "corner" or "angle", in analogy to other polygons. Since "gon" means "angle", it is analogously called a quadrangle, or 4-angle. A quadrilateral with vertices , , and is sometimes denoted as .

<span class="mw-page-title-main">Triangle</span> Shape with three sides

A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero-dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of angles of a triangle always equals a straight angle. The triangle is a plane figure and its interior is a planar region. Sometimes an arbitrary edge is chosen to be the base, in which case the opposite vertex is called the apex; the shortest segment between base and apex is the height. The area of a triangle equals one half the product of height and base length.

<span class="mw-page-title-main">Parallelogram</span> Quadrilateral with two pairs of parallel sides

In Euclidean geometry, a parallelogram is a simple (non-self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations.

<span class="mw-page-title-main">Incircle and excircles</span> Circles tangent to all three sides of a triangle

In geometry, the incircle or inscribed circle of a triangle is the largest circle that can be contained in the triangle; it touches the three sides. The center of the incircle is a triangle center called the triangle's incenter.

<span class="mw-page-title-main">Cyclic quadrilateral</span> Quadrilateral whose vertices can all fall on a single circle

In Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. The center of the circle and its radius are called the circumcenter and the circumradius respectively. Other names for these quadrilaterals are concyclic quadrilateral and chordal quadrilateral, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

<span class="mw-page-title-main">Trapezoid</span> Convex quadrilateral with at least one pair of parallel sides

In geometry, a trapezoid in North American English, or trapezium in British English, is a quadrilateral that has one pair of parallel sides.

<span class="mw-page-title-main">Thales's theorem</span> Angle formed by a point on a circle and the 2 ends of a diameter is a right angle

In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ABC is a right angle. Thales's theorem is a special case of the inscribed angle theorem and is mentioned and proved as part of the 31st proposition in the third book of Euclid's Elements. It is generally attributed to Thales of Miletus, but it is sometimes attributed to Pythagoras.

<span class="mw-page-title-main">Incenter</span> Center of the inscribed circle of a triangle

In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle.

<span class="mw-page-title-main">Symmedian</span> Reflection of a triangle vertexs median over its angle bisector

In geometry, symmedians are three particular lines associated with every triangle. They are constructed by taking a median of the triangle, and reflecting the line over the corresponding angle bisector. The angle formed by the symmedian and the angle bisector has the same measure as the angle between the median and the angle bisector, but it is on the other side of the angle bisector.

<span class="mw-page-title-main">Isosceles trapezoid</span> Trapezoid symmetrical about an axis

In Euclidean geometry, an isosceles trapezoid is a convex quadrilateral with a line of symmetry bisecting one pair of opposite sides. It is a special case of a trapezoid. Alternatively, it can be defined as a trapezoid in which both legs and both base angles are of equal measure, or as a trapezoid whose diagonals have equal length. Note that a non-rectangular parallelogram is not an isosceles trapezoid because of the second condition, or because it has no line of symmetry. In any isosceles trapezoid, two opposite sides are parallel, and the two other sides are of equal length, and the diagonals have equal length. The base angles of an isosceles trapezoid are equal in measure.

<span class="mw-page-title-main">Ptolemy's theorem</span> Relates the 4 sides and 2 diagonals of a quadrilateral with vertices on a common circle

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral. The theorem is named after the Greek astronomer and mathematician Ptolemy. Ptolemy used the theorem as an aid to creating his table of chords, a trigonometric table that he applied to astronomy.

In geometry, a cevian is a line segment which joins a vertex of a triangle to a point on the opposite side of the triangle. Medians and angle bisectors are special cases of cevians. The name "cevian" comes from the Italian mathematician Giovanni Ceva, who proved a well-known theorem about cevians which also bears his name.

<span class="mw-page-title-main">Tangential quadrilateral</span> Polygon whose four sides all touch a circle

In Euclidean geometry, a tangential quadrilateral or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral. This circle is called the incircle of the quadrilateral or its inscribed circle, its center is the incenter and its radius is called the inradius. Since these quadrilaterals can be drawn surrounding or circumscribing their incircles, they have also been called circumscribable quadrilaterals, circumscribing quadrilaterals, and circumscriptible quadrilaterals. Tangential quadrilaterals are a special case of tangential polygons.

<span class="mw-page-title-main">Law of cosines</span> Property of all triangles on a Euclidean plane

In trigonometry, the law of cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides and opposite respective angles and , the law of cosines states:

<span class="mw-page-title-main">Bicentric quadrilateral</span> Convex, 4-sided shape with an incircle and a circumcircle

In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral.

<span class="mw-page-title-main">Pythagorean theorem</span> Relation between sides of a right triangle

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

<span class="mw-page-title-main">Ex-tangential quadrilateral</span> Convex 4-sided polygon whose sidelines are all tangent to an outside circle

In Euclidean geometry, an ex-tangential quadrilateral is a convex quadrilateral where the extensions of all four sides are tangent to a circle outside the quadrilateral. It has also been called an exscriptible quadrilateral. The circle is called its excircle, its radius the exradius and its center the excenter. The excenter lies at the intersection of six angle bisectors. These are the internal angle bisectors at two opposite vertex angles, the external angle bisectors at the other two vertex angles, and the external angle bisectors at the angles formed where the extensions of opposite sides intersect. The ex-tangential quadrilateral is closely related to the tangential quadrilateral.

<span class="mw-page-title-main">Newton–Gauss line</span> Line joining midpoints of a complete quadrilaterals 3 diagonals

In geometry, the Newton–Gauss line is the line joining the midpoints of the three diagonals of a complete quadrilateral.

<span class="mw-page-title-main">Midpoint theorem (triangle)</span> Geometric theorem involving midpoints on a triangle

The midpoint theorem or midline theorem states that if the midpoints of two sides of a triangle are connected, then the resulting line segment will be parallel to the third side and have half of its length. The midpoint theorem generalizes to the intercept theorem, where rather than using midpoints, both sides are partitioned in the same ratio.

References

  1. Alfred S. Posamentier: Advanced Euclidean Geometry: Excursions for Students and Teachers. Springer, 2002, ISBN   9781930190856, pp. 3-4
  2. Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN   978-0-486-46237-0, p. 149 (original publication 1929 with Houghton Mifflin Company (Boston) as Modern Geometry).
  3. Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications.
    (3 vols.): ISBN   0-486-60088-2 (vol. 1), ISBN   0-486-60089-0 (vol. 2), ISBN   0-486-60090-4 (vol. 3). Heath's authoritative translation plus extensive historical research and detailed commentary throughout the text.

Further reading