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In a triangle,what is the ratio of the distance between a vertex and the orthocenter and the distance of the circumcenter from the side opposite vertex.

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  • $\begingroup$ Note if you connect the midpoints of the sides of the triangle, you'd have a similar triangle that its orthocenter is circumcenter of the main triangle $\endgroup$ Jan 15, 2017 at 14:34

2 Answers 2

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The triangle is $\Delta ABC$. Let's say that $O$ is the circumcenter, $H$ the orthocenter and $G$ is the centroid. We are going to use a very usefull result that comes from the Euler Line. The Euler Line says that $OH=3GO \quad (1)$.

If we use vectors to solve that problem the relation $(1)$ says that if we place the origin at $O$ then the coordinates of the orthocenter will be given by $H=A+B+C$ and then:

$$\vec{AH}=H-A=B+C$$

We also know that the distance from the circuncenter ($O$) to the side ($BC$) is the distance from $O$ to the midpoint (called $M$) of $BC$.

$$\vec{OM}=M=\frac{B+C}{2}$$

Then

$$\vec{AH}=2\cdot \vec{OM}\Leftrightarrow\frac{|\vec{AH}|}{|\vec{OM}|}=2$$

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Hint: If $A$ is the vertex, $A_1$ is the midpoint of the opposite side, $O$ is the orthocenter, $C$ is the circumcenter and $M$ is the centroid, just check the similar triangles $AOM$ and $A_1CM$.

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  • $\begingroup$ still didnt get it $\endgroup$
    – Pole_Star
    Jan 15, 2017 at 15:21
  • $\begingroup$ can u solve it.. $\endgroup$
    – Pole_Star
    Jan 15, 2017 at 15:21
  • $\begingroup$ Can you see why are they similar triangles? By their similarity $\frac{A_1C}{AO}=\frac{A_1M}{AM}=\frac{1}{2}$. $\endgroup$
    – Sz_Z
    Jan 15, 2017 at 15:28
  • $\begingroup$ i hope u get notifications $\endgroup$
    – Pole_Star
    Jan 15, 2017 at 15:28

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