Angle Bisector Theorem

Angle bisector theorem is one of the most important theorems in geometry. The basic knowledge of a few terminologies is required to understand the angle bisector theorem. An angle is a separation between any two line segments. Bisect in English means to divide into two equal halves. An angle bisector is a line that divides an angle into two congruent parts. If the angle of separation between the two lines is 900, then the two lines are said to be perpendicular to each other. The perpendicular bisector of a line is that line which is perpendicular to the given line and divides the line into two identical halves.

Angle Bisector Theorem Statement:

The angle bisector theorem establishes a relationship between the lengths of the 2 sides of a triangle and the line segments formed when the angle bisector of the angle opposite to the third side is extended to meet the third side in a triangle. 

The Angle Bisector theorem statement is as follows. “ In any triangle, when the angle bisector of one angle of a triangle is extended to meet the side opposite to that angle, the angle bisector divides the opposite sides in such a way that they are proportional to the other two sides of the triangle”. In other words, the angle bisector theorem statement can be rephrased as ‘if a point is on the angle bisector of an angle in a triangle, then the point is equidistant from the sides of the angle’.

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Explanation of what is Angle Bisector Theorem Converse:

In a triangle ABC as shown in the figure above, AD is the angle bisector of the angle A touching the side BC opposite to angle A. Then, according to the angle bisector theorem, AD divides BC in such a way that the quotient of BD and DC is equal to the quotient of AB and AC. It is written mathematically as:

\[\frac{BD}{DC} = \frac{AB}{AC}\] when∠CAD

Converse of Angle Bisector Theorem Statement:

The converse statement of the angle bisector theorem says that if in a triangle ABC, D is a point on BC such that the ratio of BD and CD and that of AB and AC are the same, then AD is the angle bisector of A.

In other words, if an interior point is equidistant from the two sides of a triangle, then the point lies on the angle bisector of the angle formed by these two lines in any triangle. 

Explanation:

If D is a point in the interior of the triangle ABC such that it is at the same distance from the sides AB and AC, then the point D lies on the angle bisector of the angle A.

Let D be а роint оn the line BС, nоt equal tо B оr С аnd such thаt АD is nоt аn аltitude of triangle АBС.

Let B1 be the bаse оf the аltitude in the triаngle АBD thrоugh B аnd let С1 be the bаse оf the аltitude in the triаngle АСD thrоugh С. Then, if D is striсtly between B аnd С, оne аnd only one оf B1 оr С1 lies inside triangle АBС and it саn be without lоss оf generality thаt B1 dоes. This саse is deрiсted in the аdjасent diаgrаm. If D lies оutside оf segment BС, then neither B1 nоr С1 lies inside the triаngle.

∠ DB1B and ∠ DС1С are right angles, while the аngles ∠ B1DB аnd ∠ С1DС аre соngruent if D lies on the segment BС (thаt is, between B аnd С) аnd they аre identiсаl in the оther саses being соnsidered, sо the triangles DB1B and DС1С аre similar (ААА), whiсh henсe рrоves the аngel biseсtоr theоrem.

Аs рer the Аngle Biseсtоr theоrem, the angle biseсtоr оf а triangle bisects the орроsite side in such а wау thаt the rаtiо оf the twо line-segments is рrороrtiоnаl tо the rаtiо оf the other two sides. Thus the relative lengths оf the орроsite side (divided by аngle biseсtоr) аre equated tо the lengths of the other twо sides оf the triаngle. This theоrem is аррliсаble tо аll tyрes оf triаngles. 

Аn аngle biseсtоr is а strаight line drаwn frоm the vertex оf а triаngle to its орроsite side in such а wау, thаt it divides the аngle intо twо equаl оr соngruent аngles. Nоw let us see, whаt is the аngle biseсtоr theоrem. In а triаngle, if the interior point is equidistant from the two sides of the triangle then that роint lies on the angle biseсtоr оf the angle formed by the twо line segments.

What is the History of the Angle Bisector Theorem?

The hypothesis of a double point is seen as Proposition 3 for Book VI in Euclid's Elements. According to Heath, a comparative explanation for the double point was given by Robert Simson who realized that Pappus expected this result without certainty. Heath goes on to say that Augustus De Morgan suggested that the two beliefs should be grouped as follows: 

Assuming that the point of a triangle is divided into or by a straight line that cuts the opposite side or the opposite side is introduced, the pieces of that side will have the same value as the different sides of the triangle; and, assuming that the side of the triangle is divided internally or remotely so that its parts have the same value as the different sides of the triangle, a straight line drawn from the area to the opposite rakish area on the main pointed side will be cut. up inside or outside the area at that specific point.

What is the Application of the Angle Bisector Theorem?

The incenter of a triangle is the centroid of the triangle and is a point defined for the triangle in a way that is independent of the position or size of the triangle. The centroid can be defined as the intersection of bisectors of the interior side of the triangle, the point equidistant from the sides of the triangle, and the intersection of the central axis and the innermost transform point. A burnt triangle and the centre of the triangle's inscribed circle. 

The intersection of the bisectors of the three vertices of triangle ABC is the centroid. The inscribed circle touches each side of the triangle. 

It is one of the four centres of the triangle known to the ancient Greeks, with the centroid, the circumcenter, and the orthogonal centre, and it is the only one not found

How is the Converse of the Angle Bisector Theorem Applied If the Point is in the Exterior of the Angle?

The converse of this theorem is likewise real. Angle Bisector Theorem Converse: If a factor is with inside the indoors of a perspective and equidistant from the sides, then it lies at the bisector of the perspective. Because the Angle Bisector Theorem and its communication are each real we have a No biconditional statement.

How to State and Prove Angle Bisector Theorem:

Angle bisector theorem is obtained by the law of sine or Sine rule in trigonometry. 

Angle Bisector Theorem Statement:

The angle bisector of one angle of a triangle divides the side opposite to it at a particular point such that the ratio in which the side is divided is equal to the ratio of the other two sides of the triangle.

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Given Data to State and Prove Angle Bisector Theorem:

ABC is a triangle with sides AB, BC, and AC. The angle bisector of A is AD which divides BC into BD and CD.

To Prove:

\[\frac{BD}{CD} = \frac{AB}{AC}\]

Angle Bisector Theorem Proof:

This is the easiest method to arrive at the Angle bisector theorem proof. However, there are a few other complex methods to understand how to state and prove angle bisector theorems.

Angle Bisector Theorem Examples:

If in a triangle ABC, AD is the angular bisector of ∠A which touches the side BC at D. Find AB and AC such that BD = 2 cm, CD = 5 cm, and AB + AC = 10 cm.

Solution:

The above problem is one of the angle bisector theorem examples. Applying angle bisector theorem to triangle ABC, we get 

\[\frac{BD}{CD} = \frac{AB}{AC}\]

It is given that BD = 2 cm and CD = 5 cm. If AB + AC = 10, then AC = 10 - AB. Substituting the values of AC, BD and CD in the above equation, we get 

\[\frac{2}{3} = \frac{AB}{10 − AB}\]

2 (10 - AB) = 5 AB

20 - 2 AB = 5 AB

20 = 5 AB + 2 AB

7 AB = 20 

AB = 20/7 cm

AC = 10 - AB

AC = 10 - 20/7

AC = 50/7 cm

Fun Facts:

Problems involving angle bisector theorem examples demand the basic knowledge of ratio and proportions.

Apart from using the sine rule, there are various other approaches to arrive at the Angle bisector theorem proof.