locus of circumcenter is a straight line

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0 replies

jlacosta

Apr 2, 2025

0 replies

Circumcenter lie on a line.

RootofUnityfilter 0

6 minutes ago

Source: my teacher

Let $ABC$ be a triangle with circumcircle $(O)$ , incentre $I$ . The second intersections of $AI$ , $BI$ , and $CI$ with $(O)$ are $X_A$ , $X_B$ , and $X_C$ , respectively. Lines $AI$ and $BC$ intersect at $D$ and lines $BX_C$ and $CX_B$ intersect at $X$ . Suppose $(XX_BX_C)$ and $(XBC)$ intersect again at $S \ne X$ . Lines $BX$ and $CX$ intersect the circumcircle of triangle $SXX_A$ again at $P \ne X$ and $Q \ne X$ , respectively.
Prove that the circumcentre of triangle $SID$ lies on $PQ$ .

0 replies

RootofUnityfilter

6 minutes ago

0 replies

ab + c + d = 3, bc + d + a = 5, cd + a + b = 2, da + b + c = 6

parmenides51 9

N 9 minutes ago by DensSv

Source: 2021 JBMO TST Bosnia and Herzegovina P1

Determine all real numbers $a, b, c, d$ for which
$ab + c + d = 3$ $bc + d + a = 5$ $cd + a + b = 2$ $da + b + c = 6$

9 replies

parmenides51

Oct 7, 2022

DensSv

9 minutes ago

Prime number and composite number

mingzhehu 3

N 9 minutes ago by mingzhehu

I have one topic on how to identify Prime Number and Composite Number quickly? Maybe the number is more than 100 or 1000.......!
If there are some formula that can be used to verify the number easily, it will be highly appreciated.
Does anybody has any good idea for that?

3 replies

mingzhehu

Today at 8:06 AM

mingzhehu

9 minutes ago

Problems for v_p(n)

xytunghoanh 1

N 26 minutes ago by whwlqkd

Hello everyone. I need some easy problems for $v_p(n)$ (use in a problem for junior) to practice. Can anyone share to me?
Thanks :>

1 reply

1 viewing

xytunghoanh

an hour ago

whwlqkd

26 minutes ago

No more topics!

locus of circumcenter is a straight line

parmenides51 1

N Feb 5, 2020 by ricarlos

Source: U sviti matematyky / In the World of Mathematics 426

Triangle $ABC$ is given. Point $M$ moves along the side $BA$ , and point $N$ moves along the extension of the side $AC$ after point $C$ in such a way that $BM = CN$ . Prove that the circumcenter of triangle AMN moves along a straight line.

(V. Yasinskyy, Vinnytsya)

1 reply

parmenides51

Feb 2, 2020

ricarlos

Feb 5, 2020

locus of circumcenter is a straight line

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Source: U sviti matematyky / In the World of Mathematics 426

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parmenides51

30629 posts

parmenides51

#1 h Feb 2, 2020, 5:46 PM • 2 Y

Y by Adventure10, Mango247

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ricarlos

255 posts

ricarlos

#2 h Feb 5, 2020, 1:24 AM • 1 Y

Y by Adventure10

Let $L$ be the midpoint of arc $BC\neq CAB$ of circumcircle of $ABC$ .
$ABLC$ is cyclic then $\angle MBL = \angle NCL$ , $BL=CL$ and $BM=CN$ then $\Delta MBL\cong \Delta NCL$
So $ML=NL\rightarrow MLN$ is an isosceles.
$\angle BLC = 180-A$ ,
$\angle MLN =\angle BLC -\angle MLB +\angle NLC =\angle BLC = 180-A$ .
Then $AMLN$ is cyclic, $AL$ is a chord of $(AMLN)$ and is constant, then the circumcenter of $(AMN)=(AMLN)$ is on the perpendicular bisector of $AL$ .

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