# Important theorems of Geometry(Triangles) – 1

Today we will be discussing about  2 important Triangle theorems which have very High applications in field of Geometry

1) Angle Bisector theorem
2) Stewart’s Theorem

Angle bisector theorem :

Introduction :

The Angle Bisector Theorem states that given triangle  and angle bisector AD, where D is on side BC, then c/m= b/n . Likewise, the converse of this theorem holds as well.

Proof :

Because of the ratios and equal angles in the theorem, we think of similar triangles. There are not any similar triangles in the figure as it now stands, however. So, we think to draw in a carefully chosen line or two. Extending AD until it hits the line through C parallel to AB does just the trick:

Since AB and CE are parallel, we know that  and . Triangle ACE is isosceles, with AC = CE.

By AA similarity. By the properties of similar triangles, we arrive at our desired result:

c/m = b/n

Stewarts Theorem :

Introduction :

Given a triangle  with sides of length  opposite vertices , respectively. If cevian  is drawn so that  and , we have that . (This is also often written , a form which invites mnemonic memorization, e.g. “A man and his dad put a bomb in the sink.”)

Proof :

Applying the Law of Cosines in triangle  at angle  and in triangle  at angle , we get the equations

Because angles  and  are supplementary. We can therefore solve both equations for the cosine term. Using the trigonometric identity   gives us

Setting the two left-hand sides equal and clearing denominators, we arrive at the equation: . However,  so and we can rewrite this as  (A man and his dad put a bomb in the sink).
Thank you

Source :
Wikepedia
Art of problem solving

I am a young student who loves math . I like number theory and inequalities part the most , and preparing for Math Olympiads :)

Posted on Monday,May 28, 2012, in Geometry and tagged , , , , , , , , , , . Bookmark the permalink. 9 Comments.

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6. Dinesh Chavan

I have 5 proofs of my own for angle biscetor theorem ,,,,instead if doing construction ,, You may apply STEWARD’s THEOREM for proving the resulttry and if you dont get it you may ask for it from me…Instead of all this mess,,you can also prove all this upto 10 std knowledge

7. dedusuiu

a

8. dedusuiu

and here we have as say prof dr mircea orasanu some important aspects of
STEWART THEOREM