#7) Concurrency relating to Squares on sides of a Triangle

by BartSimpsons, Feb 11, 2018, 2:17 PM

The following problem was discovered by pro_4_ever some time ago.

Problem. In $\triangle{ABC}$ let $BCA_cA_b$ , $CAB_aB_c$ and $ABC_bC_a$ be squares erected externally on the sides $BC,CA$ and $AB$ respectively. Let $X_a$ be the midpoint of $C_bB_c$ and define $X_b$ and $X_c$ similarly. Prove that $AX_a,BX_b$ and $CX_c$ are concurrent.

Solution

Here's more information on Jacobi's Theorem

This post has been edited 4 times. Last edited by pro_4_ever, Mar 1, 2018, 1:34 PM

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Complex Kills It!

The fact that $\Delta BX_aC$ is an isosceles right triangle, becomes trivial by complex numbers.

Let $p$ denote the complex number of point $P$ .

Now assume $\Delta ABC$ is oriented counter-clockwise.

So, rotating $A$ about $C$ by $90^\circ$ clockwise gives $B_c$ . That is $b_c= a + i(a-c)$
Similarly, noting that rotation of $A$ about $B$ by $90^\circ$ counterclockwise gives $C_b$ gives $c_b = b - i(a-b)$
Hence, the mid point $X_a$ of $B_cC_b$ is $\frac{b+c}2 + i\cdot \frac{b-c}2$
which is nothing but the third vertex of one of the isosecles right triangle, with base $BC$ .

Hence, $\Delta BX_aC$ is isosceles and right.

$\blacksquare$

by pro_4_ever, Feb 11, 2018, 2:40 PM

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Hey, what a nice blog!!!!!
by Power_Set, Jun 24, 2018, 9:53 PM
Is your blog dead ? No new post in almost 2 months.

My blog has died due to malnutrition of problems.
by integrated_JRC, May 2, 2018, 5:52 AM
@ below, I will do it soon
by pro_4_ever, Mar 1, 2018, 10:37 AM
Hi, I think we should 'hide' our solutions, as it is difficult to go up and down the blog
by Drunken_Master, Mar 1, 2018, 10:36 AM
Yeah, I created a test blog, it seems to be an AoPS problem not CSS, can't do anything about it. I'll post it in site support forum later.
by Vrangr, Feb 20, 2018, 5:48 AM
The problem is not in the CSS, I believe, as the CSS only looks after the appearance of the blog, and not the $\text{\LaTeX}$ rendering.
by WizardMath, Feb 19, 2018, 11:34 PM
I'll fix the CSS after boards of need be (I'll probably be starting my own blog too then)
by Vrangr, Feb 19, 2018, 4:41 PM
@ below,
Pls ask WizardMath...
The CSS is taken from his amazing Blog, "An Olympiad Journey"...
I know nothing abt programming.
(BTW I got permission for taking the CSS. No Copyright Issues!)
by pro_4_ever, Feb 19, 2018, 4:16 PM
Huge flaw in website design: comment gets erased if while typing $\LaTeX$ has an error and needs to be fixed.
by Vrangr, Feb 19, 2018, 12:26 PM
Upon WizardMath's Request, let us all post synthetic solutions. Well, let's declare that this blog is intended to give the reader some good problems with simple synthetic solutions. All the best. Not trying to be rude,but I might delete upcoming Bashes
by pro_4_ever, Feb 13, 2018, 4:50 PM
Here's an idea. Ban all bashes on this blog from now on. Also try posting your own previous geo solutions as storage (I hope this is allowed). This would make the quality of the blog better. Bashes are basically write-once, read-never things. Why bother?
by WizardMath, Feb 13, 2018, 4:33 PM
Somebody type Egmo 2013 Problem 1.
No bash,purely troll problem.
by QWERTYphysics, Feb 13, 2018, 4:11 PM
I hate having to see a pure geometry blog being swamped to death with bashes.

Obviously a bash is a super terrible idea. Don't do one.
Unless you are bad at synthetic.
by WizardMath, Feb 13, 2018, 2:58 PM
@ayan Done:)
by pro_4_ever, Feb 13, 2018, 1:38 PM
9th shout
by MEGAKNIGHT, Feb 13, 2018, 11:23 AM
@below Pls don't post a bash if you are thinking about posting a new entry!
by pro_4_ever, Feb 13, 2018, 11:11 AM
Wait, I thought bash is frowned upon in this blog.
by WizardMath, Feb 12, 2018, 6:57 PM
Hmmm...
Combi Geometry is accepted...
by pro_4_ever, Feb 10, 2018, 7:32 AM
Can we post Combi here?
by ayan.nmath, Feb 10, 2018, 7:31 AM
Parag dey
by Paragdey12, Feb 6, 2018, 1:17 PM
More posts please
by AnArtist, Feb 6, 2018, 2:38 AM
Second Shout!
by ccx09, Feb 3, 2018, 9:20 PM
1st shout!
by AnArtist, Jan 31, 2018, 9:20 AM

23 shouts

Problems In Geometry

#7) Concurrency relating to Squares on sides of a Triangle

by BartSimpsons, Feb 11, 2018, 2:17 PM

1 Comment