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Two lines $GE$ and $FD$ meet in $A$. They cut each other at $45$ degrees. Both $G$ and $F$ lie on the circumcircle of square $ALBK$ such that $E$ and $D$ lie on the $KB$ and $LB$ respectively without lying on the corners of the square. I'm supposed to prove that $GF$ and $ED$ are parallel. 1) How does one prove parallelism of lines in general? 2) $45$ degrees seems rather specific, could the same be proven with any angle? (i.e. $20$ degrees)

  • Are they parallel? Take the particular case where $D=B$; then $F=B$, too, so that (still distinct) lines $GF$ and $EB$ meet. – Blue Dec 11 '18 at 08:58
  • Yes, the are parallel. However I have no clue how to show this. –  Dec 11 '18 at 10:07
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    Now that you've changed the relevant angle from $30^\circ$ to $45^\circ$, yes: the lines are parallel. (My counterexample fails, because, when $D=B=F$, we have $G=E=K$, so that the lines are no longer distinct; their meeting at $B$ is no longer problematic. In any case, the counterexample answers your second question.) – Blue Dec 11 '18 at 10:27
  • I'm sorry for this inconvenience. –  Dec 11 '18 at 10:37
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    This question belongs an ongoing competition! (see the translated exercises here: https://math.meta.stackexchange.com/questions/29448/german-contest-bundeswettbewerb-mathematik?cb=1) – Dr. Mathva Dec 11 '18 at 19:08
  • I've flagged this for moderator attention. – Blue Dec 11 '18 at 19:17

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