# 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 $p^2\leq m_a^2+m_b^2+m_c^2.$

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