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Given two congruent triangles that are not a rotation, translation or reflection of each other; how can I find the glide reflection (the last remaining option) using only compass and straightedge.

Find the Glide Line and Glide Vector gives an algebraic solution but I would like a solution with a compass or straightedge.

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In this diagram, We want to glide the triangle $\triangle ABC$ to $\triangle A'B'C'$. They must meet your requirements. Of course, under those requirements they will also have opposite orientations.

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Under your requirements, at least one pair of corresponding sides of the triangles will not be parallel. In the diagram, I used $\overline{BC}$ and $\overline{B'C'}$. Extend those sides (to lines $h$ and $j$) until they intersect, at point $D$.

Those intersecting lines at $D$ have two angle bisectors. Construct the one (called $l$) such that the direction of vector $\overrightarrow{BC}$ relative to point $D$ is opposite that of vector $\overrightarrow{B'C'}$ relative to point $D$. I.e. the orientation of vector $\overrightarrow{BC}$ after reflection in $l$ would match the orientation of vector $\overrightarrow{B'C'}$.

(If somehow my analysis that there would be a pair of intersecting extended sides is false, just choose a pair of sides that are parallel [and not anti-parallel] and extend one of those sides into line $l$.)

Now construct the line $m$ through point $B'$ and perpendicular to line $l$, then line $n$ through point $B$ parallel to line $l$. Lines $m$ and $n$ are perpendicular: call their intersection point $B'_1$. Construct point $F$ as the midpoint of segment $\overline{B'B'_1}$.

Finally, construct line $p$ through point $F$ perpendicular to line $m$. This is the line of reflection for your glide transformation. Construct vector $\overrightarrow{BB'_1}$, the translation vector for your glide transformation.

To show this is correct, I have shown the translation of $\triangle ABC$ under vector $\overrightarrow{BB'_1}$. We see that $\triangle A'B'C'$ is indeed the reflection of that triangle in line $p$.

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