Important Results in Geometry

Theorem 1
 
Let \( M \) be a point on the side \( BC \) of the triangle \( ABC \). Then \[ \overrightarrow{AM}=\frac{\overrightarrow {MC}}{\overrightarrow {BC}}\cdot \overrightarrow {AB}+\frac{\overrightarrow {BM}}{\overrightarrow {BC}}\cdot \overrightarrow {AC}.\]

In the sequel, we will write \( BM \) as the length of the vector \( \overrightarrow{BM} \). The result of the previous theorem can be also expressed as \[ \overrightarrow{AM}=\frac{{MC}}{ {BC}}\cdot \overrightarrow {AB}+\frac{ {BM}}{ {BC}}\cdot \overrightarrow {AC}.\]

Theorem 2 (Stewart’s theorem)
 
Let \( M \) be a point on the side \( BC \) of the triangle \( ABC \). Then the following equality holds: \[ AM^2=\frac{MC}{BC}\cdot AB^2+\frac{BM}{BC}\cdot AC^2-BM\cdot MC.\]

Problem 1
 
Let \( ABC \) be a triangle, \( a \), \( b \), \( c \) its side-lengths, and \( m_a \) the length of the median corresponding to the side \( a \). Prove that \[ m_a^2=\frac{2b^2+2c^2-a^2}4.\]

Problem 2
 

Let \( ABC \) be a triangle whose side lengths are \( AB=c \), \( BC=a \), \( CA=b \). Denote by \( A^{\prime\prime} \) the point where the internal bisector intersects the side \( BC \). Let \( p \) be the semi-perimeter of the triangle \( ABC \).

  • (a) Prove that \( BA^{\prime\prime}:A^{\prime\prime}C=AB:AC \).

  • (b) Prove that \( l_a=AA^{\prime\prime}=\frac{2}{b+c}\cdot \sqrt{bcp(p-a)} \).

Problem 3
 

Let \( O \) be the circumcenter of the triangle \( ABC \) and \( G \) its centroid. Prove that \( OG^2=R^2-\frac19(a^2+b^2+c^2) \).

Problem 4
 

If \( a \), \( b \), \( c \) are the lengths of the sides of \( \triangle ABC \) and \( R \) its circumradius prove that \[ 9R^2\geq a^2+b^2+c^2.\]

Theorem 3 (Incircle-excircle theorem)
 

Let \( ABC \) be the given triangle and let \( O \) be its circumcenter, \( I \) its incenter, \( I_a \), \( I_b \), \( I_c \) the centers of the excircles \( k_a \), \( k_b \), \( k_c \) (corresponding to the sides \( BC \), \( CA \), \( AB \)), and \( G \) its centroid. Denote by \( a \), \( b \), \( c \) the side lengths, \( R \) the circumradius, and by \( r \) the inradius or \( \triangle ABC \). Denote by\( r_a \), \( r_b \), \( r_c \) be the exradii. Let \( p \) be the semi-perimeter of \( \triangle ABC \). Then the following statements hold:

  • (a) \( AI \) intersects the circumcircle at the midpoint \( Q \) of the arc \( BC \). \( I_bI_c \) contains the point \( A \) and the midpoint \( P \) of the arc \( BC \) that contains \( A \). The point \( O \) belongs to \( PQ \).

  • (b) If \( M \) and \( N \) are the points of tangency of \( k_b \) and \( k_c \) with \( BC \), then \( P \) is the midpoint of \( I_aI_b \), \( A_1 \) is the midpoint of \( MN \) and \( PA_1=\frac{r_b+r_c}2 \).

  • (c) Denote by \( U \) the point of tangency of the incircle with \( BC \) and by \( V \) the point of tangency of \( k_a \) with \( BC \). Then \( A_1 \) is the midpoint of \( UV \), \( Q \) is the midpoint of \( II_a \), and \( QA_1=\frac{r_a-r}2 \).

  • (d) (Heron’s formula) \( S_{ABC}=\sqrt{p(p-a)(p-b)(p-c)} \).

Problem 5
 

Prove that \[ p^2\leq m_a^2+m_b^2+m_c^2.\]


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us