Application of Basic Proportionality Theorem

Here we will prove that the internal bisector of an angle of a triangle divides the opposite side in the ratio of the sides containing the angle.

Given: XP is the internal bisector of ∠YXZ, intersecting YZ at P.

Application of Basic Proportionality Theorem

To prove: \(\frac{YP}{PZ}\) = \(\frac{XY}{XZ}\).

Construction: Draw ZQ  ∥ XP such that ZQ meets YX produced at Q.



1. ∠YXP = ∠XQZ

2. ∠PXZ = ∠XZQ

3. ∠XQZ = ∠XZQ

4. XQ = XZ

5. \(\frac{YX}{XQ}\) = \(\frac{YP}{PZ}\)

6. \(\frac{YX}{XZ}\) = \(\frac{YP}{PZ}\)


1. XP  ∥ QZ and YQ is a transversal

2. XP  ∥ QZ and XZ is a transversal

3. ∠YXP = ∠PXZ

4. ∠XQZ = ∠XZQ

5. XP ∥ QZ

6. By statement 4.


1. The above proposition is true for external division also.

So, \(\frac{YP}{ZP}\) = \(\frac{XY}{XZ}\)

Application of Basic Proportionality Theorem Image

2. Converse of the above proposition is also true.

So, if P is a point on YZ such that YP : PZ = XY : XZ then XP bisects the angle YXZ internally or externally.

9th Grade Math

From Application of Basic Proportionality Theorem to HOME PAGE

Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.

Share this page: What’s this?

Recent Articles

  1. Fundamental Geometrical Concepts | Point | Line | Properties of Lines

    Apr 18, 24 02:58 AM

    Point P
    The fundamental geometrical concepts depend on three basic concepts — point, line and plane. The terms cannot be precisely defined. However, the meanings of these terms are explained through examples.

    Read More

  2. What is a Polygon? | Simple Closed Curve | Triangle | Quadrilateral

    Apr 18, 24 02:15 AM

    What is a polygon? A simple closed curve made of three or more line-segments is called a polygon. A polygon has at least three line-segments.

    Read More

  3. Simple Closed Curves | Types of Closed Curves | Collection of Curves

    Apr 18, 24 01:36 AM

    Closed Curves Examples
    In simple closed curves the shapes are closed by line-segments or by a curved line. Triangle, quadrilateral, circle, etc., are examples of closed curves.

    Read More

  4. Tangrams Math | Traditional Chinese Geometrical Puzzle | Triangles

    Apr 18, 24 12:31 AM

    Tangram is a traditional Chinese geometrical puzzle with 7 pieces (1 parallelogram, 1 square and 5 triangles) that can be arranged to match any particular design. In the given figure, it consists of o…

    Read More

  5. Time Duration |How to Calculate the Time Duration (in Hours & Minutes)

    Apr 17, 24 01:32 PM

    Duration of Time
    We will learn how to calculate the time duration in minutes and in hours. Time Duration (in minutes) Ron and Clara play badminton every evening. Yesterday, their game started at 5 : 15 p.m.

    Read More