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.
2026-03-26 09:19:45.1774516785
Euler's Line of a medial triangle
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