0

I was studying inversion in olympiad geometry from Evan Chen's EGMO book where in a footnote he mentions that inversion preserves angles between clines, where angle is defined as angle of tangents at point of intersection.

So firstly, my question is:

Prove inversion preserves angles between clines.

Ok so I started trying to prove this, but unfortunately, the only way I knew how was to make tons of cases, like angle between line through center and circle, line and line, line and circle through center and so on (in fact it might be a good counting question to figure out how many possibilities there are ;D), I was trying to use the fact that $\measuredangle OAB = -\measuredangle OB'A'$ and making tons of small messy diagrams with a lot of tangents, and I proved a couple cases, circle through center and line through center, line and line but frankly it seemed very annoying and tough, and it took me almost 1 hour to just do 3 cases, and there are some cases I am not able to do only, like circle and circle... When I was learning Homothety, everything seemed much nicer and simpler but here it seems I gotta split stuff into tons of cases and do weird angle chasing... So I am wondering if there is a nice way, or any smaller way, or infact if anyone even knows any proof whatsoever of the fact that inversion preserves angles, I would really appreciate it.

Also btw, as I am just starting to learn inversion in olympiad geometry, I would love to know of any handouts/ references that you liked.

Thank you!

Aditya_math
  • 1,863
  • The explanation an inversion behaves locally like a symmetry ; see my answer here ; intuitively : in the vicinity of its invariant circle, it is very convincing (an answer needing to have some knowledge of analysis) ; as a consequence, like an ordinary symmetry, it preserves angles in absolute value ; more interestingly (I advise you to have this view), it changes an oriented angle into its opposite. – Jean Marie Jan 01 '23 at 23:18
  • You may be as well interested by this question of mine here – Jean Marie Jan 01 '23 at 23:24
  • What is a cline? Do you mean inversion with respect to a circle centred at the origin of a punctured plane? – P. Lawrence Jan 02 '23 at 01:30
  • @P.Lawrence: A cline is a generalized circle. – joriki Jan 02 '23 at 06:10

1 Answers1

1

After posting this question on AoPS and here, I found a solution to this problem and infact it was much easier than it had seemed before, so I am posting it here, incase someone else in the future has the same question that I had.

First some notation: Here inversion about center $O$ sends cline $c$ to $c'$

$l_0$=line through $O$ $l$ arbitrary line $\omega_0$=circle passing through $O$ $\omega$=arbitrary circle

First prove three easy cases

$(l_0,l) \leftrightarrow (l_0,\omega_0)$

$(l_0,\omega) \leftrightarrow (l_0,\omega)$

$(l_0,l_0) \leftrightarrow (l_0,l_0)$

Then for any other arbitrary clines $c_1$ and $c_2$, let $l_0$ be the line through $O$ and $c_1 \cap c_2$ And now $\measuredangle (c_1,c_2) = \measuredangle (c_1,l_0)+ \measuredangle (l_0,c_2)= -\measuredangle (c'_1,l_0) - \measuredangle (l_0,c'_2) = -\measuredangle (c'_1,c'_2)$

So done.

btw, I counted it and there would be $7$ cases in total if someone wanted to prove them one by one and not just use cline notation.

Aditya_math
  • 1,863