Basic Properties of Circum-Incentral Triangle

by AlastorMoody, Jan 7, 2019, 6:46 AM

Definition: $\text{ If } M_A,M_B,M_C \text{ are circumcenters of } \Delta BIC , \Delta AIC , \Delta AIB,$ $ \text{ then, } \Delta M_AM_BM_C \text{ is the circum-incentral triangle w.r.t to } \Delta ABC$

$\text{(1) } \text{ If } M_A,M_B,M_C \text{ are circumcenters of } \Delta BIC , \Delta AIC , \Delta AIB, \text{ then, } \Delta M_AM_BM_C \text{ is the circum-incentral triangle w.r.t to } \Delta ABC$
$\text{(2) } M_A,M_B,M_C \text{ lie on circle with circumcenter of } \Delta ABC \text{ as the center }$
$\text{(3) } A,B,C \text{ are the reflections of } I \text{ over } M_BM_C, M_AM_C , M_AM_B$
$\text{(4) } I \text{ is the orthocenter of } \Delta M_AM_BM_C$
$\text{(5) Let } S_9 \text{ be the mid-point of } I \text{ and } O, \text{ then, } S_9 \text{ is the nine-point center of } \Delta M_AM_BM_C$
$\text{(6) } \text{ Homothety } \mathcal{X} \equiv 2 \text{ at } I , \mathcal{X}: M_A \rightarrow I_A \text{ sends } \Delta M_AM_BM_C \text{ to } \Delta I_AI_BI_C$
$\text{(7) } I \text{ is the orthocenter of } \Delta I_AI_BI_C$
$\text{(8) } \Delta ABC \text{ is the orthic triangle of } \Delta I_AI_BI_C$
$\text{(9) } \Delta M_AM_BM_C \text{ is the euler tiangle of } I_AI_BI_C$
$\text{(10) } O \text{ is the nine-point center of } \Delta I_AI_BI_C$
$\text{(11) Let } A',B',C' \text{ be the mid-points of } AI,BI,CI , \text{ then } \Delta A'B'C' \text{ is the orthic triangle of } \Delta M_AM_BM_C$
$\text{(12) } S_9 \text{ is the circumcenter of } \Delta A'B'C'$
$\text{(13) If } E \text{ is the circumcenter of } \Delta I_AI_BI_C , \text{ then, } IO=OE$
$\text{(14) } I,S_9,O,E \text{ are collinear }$
$\text{(15) } \left(\Delta A'B'C' \right) \text{ is the nine-point circle of } \Delta M_AM_BM_C$


Check-out some basic properties and their proofs over here
This post has been edited 4 times. Last edited by AlastorMoody, Jan 15, 2019, 7:10 PM

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wait this is actually my favorite configuration right now :D

Edit(Alastor Moody): There is a complete ocean of geometry left to study, at this moment, I am really excited and wondering how exciting can geometry get further on !!
This post has been edited 3 times. Last edited by AlastorMoody, Jan 15, 2019, 8:14 PM

by Kagebaka, Jan 8, 2019, 11:21 PM

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The only post of yours till now that I can fully appreciate :coolspeak:

Edit(AlastorMoody) Lol! I am considering this as a compliment :D
This post has been edited 2 times. Last edited by AlastorMoody, Mar 24, 2019, 12:30 PM

by Hexagrammum16, Mar 23, 2019, 10:19 AM

I'll talk about all possible non-sense :D

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    by bookstuffthanks, Jul 31, 2024, 12:05 PM

  • hello fellow moody!!

    by crazyeyemoody907, Oct 31, 2023, 1:55 AM

  • @below I wish I started earlier / didn't have to do JEE and leave oly way before I could study conics and projective stuff which I really wanted to study :( . Huh, life really sucks when u are forced due to peer pressure to read sh_t u dont want to read

    by kamatadu, Jan 3, 2023, 1:25 PM

  • Lots of good stuffs here.

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  • But even if he went to jee he could continue with this.

    Doing JEE(and completely leaving oly) seems like a insult to the oly math he knows

    by HoRI_DA_GRe8, Feb 11, 2022, 2:11 PM

  • Ohhh did he go for JEE? Good for him, bad for us :sadge:. Hmmm so that is the reason why he is inactive
    Btw @below finally everyone falls to the monopoly of JEE :) Coz IIT's are the best in India.

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  • Kukuku first shout of 2022,why did this guy left this and went for trashy JEE

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  • nice blog :)

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