sum of vectors of alitudes is zero iff ABC is equilateral

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sum of vectors of alitudes is zero iff ABC is equilateral

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#1 h Jul 26, 2021, 8:26 PM • 1 Y

Y by teomihai

Let $AA_1$ , $BB_1$ , $CC_1$ be the altitudes of triangle $ABC$ . Prove that the equality $\overrightarrow{AA_1}+ \overrightarrow{BB_1} + \overrightarrow{C C_1}= \overrightarrow{0}$ holds if and only if triangle $ABC$ is equilateral.

This post has been edited 4 times. Last edited by parmenides51, Jul 27, 2021, 1:54 PM

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#3 h Oct 4, 2022, 7:37 AM

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parmenides51 wrote:

Let $AA_1$ , $BB_1$ , $CC_1$ be the altitudes of triangle $ABC$ . Prove that the equality $\overrightarrow{AA_1}+ \overrightarrow{BB_1} + \overrightarrow{C C_1}= \overrightarrow{0}$ holds if and only if triangle $ABC$ is equilateral.

old and nice!

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#4 h Oct 4, 2022, 11:31 AM • 2 Y

Y by Mango247, Mango247

Hint:
First we prove that:L et $a,b,c$ the lengths of the sides orthic triangle $ABC$ then:

$\overrightarrow{A_1A}=\frac{1}{c+b-a}(b \overrightarrow{BA}+c \overrightarrow{CA})$

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#5 h Oct 4, 2022, 5:09 PM • 1 Y

Y by teomihai

Using Lami's theorem, it must be true that $\frac{AA_1}{\sin A} = \frac{BB_1}{\sin B}$ and analogous relations. However, $AA_1 = 2R \sin B \sin C$ , so this implies $\sin A = \sin B$ . Analogously, $\sin B = \sin C$ , so all angles of the triangle are equal.

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