I have the following problem with a comment below on the steps that I took so far.
Here is the example: Let triangle ABC be any triangle. The midpoints of the sides in Triangle ABC are labeled $A', B', C'$ of sides: $BC, CA, AB$. Let $D, K, I$ be the circumcenter, centroid and orthocenter, $D, K, I$. Let $ D', K', I'$ be circumcenter, orthocenter of triangle $ A'B'C'$.
$1.$Prove that Euler line of triangle ABC coincides with the Euler line of triangle $A'B'C'$
$2.$ Calculate $D'D/I'I$.
We know that $A′B′C′$ and $ABC$ is in a $1:2$ ratio since it is formed by the midpoints of the larger triangle called a medial triangle. The orthocenter of Triangle $ABC$ is also going to the same as the circumcenter of the $A′B′C′$.I can safely assume that the second part will be equal to 1. And I could probably use a dilation of $-1/2$ to prove that the medial triangle is the same as the larger one. Please add comments and suggestions.
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5$\begingroup$ One person, many, many questions. Please, if you're having this much trouble with geometry, see the instructor. $\endgroup$– Gerry MyersonMay 14, 2012 at 4:55
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$\begingroup$ What does -1 mean(on the left hand side next to the grey star and vertical arrows)? $\endgroup$– user31284May 15, 2012 at 1:43
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$\begingroup$ @user31284: This site works by having users vote on the content. A user can click the up arrow (upvote) if they feel "this question shows research effort; it is useful and clear", and they can click the down arrow (downvote) if they feel "this question does not show research effort; it is unclear or not useful". The number between the arrows shows the total number of upvotes, minus the total number of downvotes. These votes are part of what determines your reputation score (see here for more info). $\endgroup$– Zev ChonolesMay 15, 2012 at 2:05
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$\begingroup$ It's a "down-vote". The objective reason why this might happen is another user views this question as "unclear" or "not showing research effort". Alas, there are many subjective reasons that you might receive a downvote. I don't know when the vote happened, but if it was before you added your work, that could be the reason. $\endgroup$– The Chaz 2.0May 15, 2012 at 2:05
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$\begingroup$ user31284, you can (should) edit your comments (especially those containing your work/efforts) into the body of the question so that those viewing will see the important information all together. $\endgroup$– The Chaz 2.0May 15, 2012 at 2:13
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1 Answer
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The triangles $ABC$ and $A'B'C'$ have the same center of gravity (intersection of medians). Furthermore, you can prove that the circumcenter of $ABC$ is the orthocenter of $A'B'C'$ (draw a precise figure when in doubt). Therefore, the Euler lines of the two triangles are the same.
We can obtain $A'B'C'$ by performing an homothety of center $K$, the centroid, and ratio $-1/2$. Therefore, $DD' = 3/2KD$. Moreover $II' = 3/2 IK$. Now, in order to find $DD'/II'$ you just need $KD/IK$ and this comes from the ratios made by the points $I,K,D$ on the Euler line.
(Using notations like $I,K,D$ is really confusing. One generally notes $G$ for the center of gravity, $O$ the circumcenter and $H$ the orthocenter)
answered Dec 18, 2017 at 20:13