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Let $\triangle{ABC}$ have side lengths $\overline{AB}=3$, $\overline{BC}=5$, $\overline{CA}=7$ and let $D$ be the circumcenter of this triangle. Also, let $E$ be the incenter of $\triangle{ABD}$. $F$ is the incenter of $\triangle{ACD}$. Find the length of segment $\overline{EF}$.

This problem is begging for a diagram, but herein lies the problem. It is difficult to construct a $3-5-7$ triangle. Even more difficult to find the circumcenter and incenter of these triangles. Help is much appreciated and needed.

A Piercing Arrow
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  • The question asks for the distance between two incenters, but the title asks for the distance between a circumcenter and an incenter, is this correct? – B. Mehta Dec 22 '17 at 20:29

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HINT

If this is the main problem then here is how to construct a 3-5-7 triangle. Assume that the radius of the little white circles is one. Then

enter image description here

zoli
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