Ellipse reflection property objectively correct proof.

by qwerty123456asdfgzxcvb, Nov 30, 2024, 3:07 AM

Let $\mathcal{C}$ be a (real) conic and let $I,J$ be the circle points, define the foci $F_1,F_2$ as one pair of points made by intersections of tangents from $I,J$ to $\mathcal{C}$ (There are two pairs, pick the pair with both intersection points real).

https://i.ibb.co/wrGpsDf/image.png

Now consider DDIT on the quadrilateral $F_1IF_2J$. There is an involution fixing the tangent at $P$ (call it $\ell$), that swaps $PF_1, PF_2$ and $PI, PJ$. So we have \[(\ell,PF_1; PI, PJ) = (\ell, PF_2; PJ, PI) = \frac{1}{(\ell, PF_2; PI, PJ)}\].
Recall the definition of the angle between two lines $\ell_1, \ell_2$ as $\frac{1}{2i}\ln(\ell_1, \ell_2; \overline{\ell_1 \cap \ell_2 I}, \overline{\ell_1 \cap \ell_2 J})$. Thus, \begin{align*} \angle \ell\overline{PF_1} &= \frac{1}{2i}\ln(\ell, PF_1;PI, PJ) \\ &= \frac{1}{2i}\ln\left( \frac{1}{(\ell, PF_2; PI, PJ)}\right) \\ &= -\frac{1}{2i}\ln(\ell, PF_2;PI, PJ)  = -\angle \ell\overline{PF_2}. \end{align*}
This post has been edited 2 times. Last edited by OronSH, Nov 30, 2024, 3:15 AM

Comment

J
U VIEW ATTACHMENTS T PREVIEW J CLOSE PREVIEW rREFRESH
J

1 Comment

The post below has been deleted. Click to close.
This post has been deleted. Click here to see post.
oh yeah this also proves the fact that for an ellipse with foci F1, F2 and a point P with tangent lines L1, L2, that PF1 and PF2 are isogonal in PL1, PL2 (since the involuion swapping PL1, PL2 swaps PF1, PF2, PI, PJ)

by qwerty123456asdfgzxcvb, Dec 5, 2024, 11:46 PM

susus

avatar

OronSH
Archives
+ August 2023
Shouts
Submit
  • how soorz

    by ChickensEatGrass, May 8, 2025, 8:39 PM

  • Hi Oron Wang!

    by Mintylemon66, May 7, 2025, 10:26 PM

  • Hi Oron Wang!

    by bjump, May 7, 2025, 4:22 PM

  • Hi Oron Wang!

    by lu1376091, May 6, 2025, 5:48 PM

  • omg im admits

    by ohiorizzler1434, May 5, 2025, 1:46 AM

  • ^
    |
    |
    ADMITS

    by CyclicISLscelesTrapezoid, Apr 30, 2025, 9:39 PM

  • admits admits admits

    by vincentwant, Apr 29, 2025, 6:41 PM

  • sus i got 134 in slope

    by bjump, Apr 29, 2025, 4:44 PM

  • oron the rizzler admits 10 quandillion aura, oron so orz

    by ohiorizzler1434, Apr 24, 2025, 12:29 AM

  • I love that I am banned :)

    by maxamc, Apr 23, 2025, 2:59 AM

  • because it was SPAM

    by PEKKA, Apr 18, 2025, 6:23 PM

  • wait why was a post by PEKKA deleted lol

    by lpieleanu, Apr 17, 2025, 7:21 PM

  • Hi! contrib pls? I literally have a math team named xooks lol

    by Cattus1221, Apr 16, 2025, 9:45 PM

  • sus_rbo.

    by sus_rbo, Apr 11, 2025, 12:06 PM

  • wsg my second favorite borosillicate glass boron isotope orzon xoron oreo

    by Inaaya, Mar 30, 2025, 10:38 PM

421 shouts
Tags
About Owner
  • Posts: 1745
  • Joined: Apr 29, 2018
Blog Stats
  • Blog created: May 16, 2023
  • Total entries: 221
  • Total visits: 37144
  • Total comments: 1179
Search Blog
a