Geometric Inequalities: Introduction

When not stated otherwise, lengths of the sides of a triangle \( ABC \) are labeled by \( a=BC \), \( b=CA \), \( c=AB \). The angles are denoted \( \alpha=\angle A=\angle BAC \), \( \beta=\angle B=\angle ABC \), \( \gamma=\angle C=\angle ACB \). The midpoints of the sides \( BC \), \( CA \), \( AB \) are denoted by \( A_1 \), \( B_1 \), \( C_1 \), and the feet of the altitudes from \( A \), \( B \), \( C \) to the opposite sides by \( A^{\prime} \), \( B^{\prime} \), and \( C^{\prime} \). We will frequently denote the points where the internal bisectors intersect the sides of the triangle by \( A^{\prime\prime} \), \( B^{\prime\prime} \), \( C^{\prime\prime} \). The lengths of the medians \( AA_1 \), \( BB_1 \), \( CC_1 \) are denoted by \( m_a \), \( m_b \), \( m_c \); the lengths of the altitudes \( AA^{\prime} \), \( BB^{\prime} \), \( CC^{\prime} \) by \( h_a \), \( h_b \), \( h_c \); and the lengths of the segments of internal bisectors by \( l_a=AA^{\prime\prime} \), \( l_b=BB^{\prime\prime} \), and \( l_c=CC^{\prime\prime} \). The semi-perimeter of the triangle will be denoted by \( p \) (i.e. \( p=(a+b+c)/2 \)). The circumradius of the triangle will be denoted by \( R \) and the inradius by \( r \). The radii of the three circles tangent to one side and the extensions of the other two sides (called the excircles) will be denoted by \( r_a \), \( r_b \), \( r_c \). \( S \) and \( S_{ABC} \) will denote the area of the triangle \( ABC \).

Theorem 1 (Triangle inequality)
 

If \( ABC \) is a triangle then the following statements hold:

  • (a) If \( a \), \( b \), \( c \) are the lengths of the sides, then \( a< b+c \), \( b< c+a \), \( c< a+b \). Conversely, if \( a \), \( b \), \( c \) are positive real numbers each of which is smaller than the sum of the other two, then there exists a triangle whose side lengths are \( a \), \( b \), and \( c \).

  • (b) \( AB< BC \) if and only if \( \angle ACB< \angle BAC \).

Problem 1
 

Prove that for arbitrary triangle the following inequalities hold: \[ p< m_a+m_b+m_c < 2p.\]

Problem 2
 

Prove that for every triangle the sum of its medians is greater than \( 3/4 \) of the sum of its sides.

Theorem 2 (Ptolemy)
 

For any four points \( A \), \( B \), \( C \), \( D \), in the plane \[ AC\cdot BD\leq AB\cdot CD+AD\cdot BC.\] The equality holds if and only if \( ABCD \) is cyclic with diagonals \( AC \) and \( BD \); or if \( A \), \( B \), \( C \), \( D \) are collinear and exactly one of \( B \), \( D \) is between \( A \) and \( C \).

Theorem 3 (Parallelogram Inequality)
 

For every four points \( A \), \( B \), \( C \), \( D \) in the space we have \[ AB^2+BC^2+CD^2+DA^2\geq AC^2+BD^2.\] The equality holds if an only if \( ABCD \) is a parallelogram (or degenerated parallelogram).

Problem 3
 

Let \( ABC \) be an acute-angled triangle. Using a straight-edge and compass construct a point \( M \) inside the triangle \( ABC \) for which the sum \( MA+MB+MC \) is minimal.

Remark. Point \( M \) obtained in the previous example is called the Toricelli point of the triangle \( ABC \). From the previous proof it follows that \( M \) is the intersection of \( AQ_A \), \( BQ_B \), and \( CQ_C \) where \( Q_A \), \( Q_B \), and \( Q_C \) are the points in the exterior of \( \triangle ABC \) for which \( \triangle BAQ_C \), \( \triangle ACQ_B \), and \( \triangle BQ_AC \) are equilateral.

Problem 4
 

Prove that \( h_a\leq l_a\leq m_a \).


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