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1 RMM-A NEW PROOF FOR BASIC PROPORTIONALITY IN TRIANGLE

A NEW PROOF OF BASIC PROPORTIONALITY

THEOREM IN TRIANGLE

By Toyesh Prakash Sharma

St. C.F. Andrews School, Agra, India e-mail:toyeshprakash@gmail.com

Abstract

In this short paper, there is a new proof of well know Basic Proportionality Theorem (BPT) in similarity of triangles have been introduced respectively in which there is no need to use the concepts of area in general method which have been used only there is need to know about sine law which basically reduce number of pages or steps in this proof respectively.

Keywords:Basic Proportionality Theorem (BPT), sine, sine law, triangle, perpendicular etc.

1 Introduction

In general, BPT i.e. Basic Proportionality Theorem is one of the basic as well as essential result in similarity of triangles for sides for it we can check [1][2] and with reference to it we can also state the statement of it which is given below-

Statement. if a line is drawn parallel to one side of a triangle to intersect the other two side in distinct points, the other two sides are divide in the same ratio.

Mathematical form. For fig 1, AD AE

BD AC

So, in section 2 we see new proof of it by using sine law.

2 Proof

In same figure fig1 we can also apply sine law in two manners so, in first we use triangle ADE with angles A, D and E same as in second manner we use triangle ABC with angles A, B and C

respectively then with reference to [3] sine law is sin sin sin

a b c

A B C if triangle is ABC with

sides a, b and c by using it for triangle ADE and triangle ABC we can rewrite it as-

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2 RMM-A NEW PROOF FOR BASIC PROPORTIONALITY IN TRIANGLE

sin sin sin

DE AE AD

A D E and

sin sin sin

BC AC AB

A B C as it is much more clear that angle ADE or D

is equal to angle B and angle AED or E is equal to C because of parallel line DE so with it we can say that

sin

sin

BC A

AC B ,

sin

sin

AC B

AB C and

sin

sin

AB C

BC A

And same as

So,

Or

With it we can

AD AE

BD AC

3 CONCLUSION

In this whole author try to proof BPT i.e. Basic Proportionality Theorem with the help of sine law which reduces the length of the proof as compare to length of general method in which there is a need of the concept of area respectively.

ACKNOWLEDGEMENT Author realize that there is much more need to understand this theorem as well as sine law that’s why he would like to do thanks to his teachers Er Manish Gupta, Mr Narendra shukla and Mr Vinay Tripathi for giving him knowledge about it.

References

[1] Textbook of Mathematics of Class-X , NCERT ch-6 p 124-125 ISBN 81-7450-6349

[2] Intercept theorem- Wikipedia.

[3] Textbook of Mathematics of Class-XI , NCERT last pages of book