# 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

Google

Posted on Monday,May 28, 2012, in Geometry and tagged angle, Angle bisector, angle bisector theorem, applications, bisectors, cosine rule, geometry, ratio, stewart;s theorem, theorems, triangles. Bookmark the permalink. 9 Comments.

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Thanks , thanks !!!!

Thanks for all the positive comments…

And I am really sorry for the late reply…

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

a

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

STEWART THEOREM