# 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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