{"id":37816,"date":"2026-01-27T10:02:05","date_gmt":"2026-01-27T10:02:05","guid":{"rendered":"https:\/\/mp.moonpreneur.com\/math-corner\/?p=37816"},"modified":"2026-02-25T09:04:42","modified_gmt":"2026-02-25T09:04:42","slug":"proof-of-angle-bisector-theorem","status":"publish","type":"post","link":"https:\/\/mp.moonpreneur.com\/math-corner\/proof-of-angle-bisector-theorem\/","title":{"rendered":"Angle Bisector Theorem: Formula, Proof, and Solved Examples"},"content":{"rendered":"\t\t
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We all have seen that one geometry question where we want a faster way to find that one missing side length. If you\u2019ve ever felt like geometry is just a collection of random rules, it\u2019s time to simplify. One of the best theorems in the math section is the Angle Bisector Theorem.<\/b> In this blog, we will see the proof and its example, too.<\/span><\/p>

What is \u201cAngle Bisector Theorem\u201d?<\/b><\/span><\/h4>

The theorem states that \u201cif you have a triangle and you draw a line that splits one of the angles exactly in half (bisecting it), that line will divide the opposite side of the triangle into two parts. The ratio of those two new segments is exactly the same as the ratio of the other two sides of the triangle<\/span><\/p>

\"\"<\/a><\/p>

Here, if triangle\u00a0 ABC is divided into two equal triangles ABD and ACD, by line AD, then the ratio \\(\\displaystyle \\frac{BD}{DC} = \\frac{AB}{AC}\\)
<\/span><\/p>

Proof of Angle Bisector Theorem<\/b><\/span><\/h4>

Statement:<\/b><\/span> The angle bisector of a triangle divides the opposite side in such a way that the ratio of the two line segments is proportional to the ratio of the other two sides.<\/span><\/p>

Let\u2019s prove this statement.<\/span><\/p>

Given<\/span>:<\/b> Let us consider triangle ABC, in which AD is the angle bisector of angle BAC, which meets BC at point D.<\/span><\/p>

To prove: \\(\\displaystyle \\frac{BD}{DC} = \\frac{AB}{AC}\\)<\/p>

Construction<\/span>:<\/b> Draw a ray CE parallel to DA. Extend BA such that it intersects CE at E<\/span>.<\/span><\/p>

\"\"<\/a><\/p>

Since CE || DA, AC acts as the transversal.<\/p>

\u2220DAC = \u2220ACE (alternate interior angles)…(1)<\/p>

\u2220BAD = \u2220AEC (corresponding angles)……..(2)<\/p>

As we know that AD is the angle bisector of\u2220BAC,<\/p>

\u2220BAD = \u2220DAC…..(3)<\/p>

From (1), (2) and (3), we have<\/p>

\u2220ACE = \u2220AEC this implies that triangle ACE is an isosceles triangle.<\/p>

Therefore, opposite sides of equal angles are equal.<\/p>

Thus,\u00a0 for triangle ACE, we can write\u00a0<\/p>

AE = AC ……..(4)<\/p>

By the triangle proportionality theorem, if a line parallel to one side of a triangle intersects the other two sides of the triangle, and then the line divides these two sides proportionally.<\/span><\/p>\\(\\displaystyle \\frac{BD}{DC} = \\frac{AB}{BE} \\qquad (5)\\)

From equation (4), we have AE = AC, so on replacing AE by AC in equation (5), we get<\/span><\/p>\\(\\displaystyle \\frac{BD}{DC} = \\frac{AB}{AC}\\)

Example of Angle Bisector Theorem<\/b><\/span><\/h4>

Example 1: Finding the Side Length<\/b><\/span><\/p>

  1. In triangle ABC, the angle bisector of angleA intersects BC at D. If AB = 6cm, AC = 9cm, and BD = 4cm, find the length of DC.<\/span><\/li><\/ol>

    \u00a0 \u00a0 \u00a0Solution: using the theorem \\(\\displaystyle \\frac{BD}{DC} = \\frac{AB}{AC}\\)
    <\/span><\/p>