# Category Archives: Geometry

## CGMO 2012 ( China Girls Math Olympiad 2012) Problem 5

As shown in the figure below, the in-circle of  is tangent to sides  and  at  and  respectively, and  is the circumcenter of . Prove that .

## CGMO 2012 (China Girls Math Olympiad 2012)- Problem 2

Circles  and  are tangent to each other externally at . Points  and  are on , lines  and  are tangent to  at  and , respectively, lines  and  meet at point
Prove that

(1) ;
(2) .

## 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