Nice problem about the Lemoine point of triangle JaBC and OI line

Given acute triangle ABC - M is a variable point on side AB. The circle circumscribing triangle BCM intersects AC at N.

a)Let I be the center of the circle circumscribing triangle AMN. Prove that I always lies on a fixed line.

b)Let J be the center of the circle circumscribing triangle MBC. Prove that line segment IJ has a constant length.

0 replies

Lukariman

20 minutes ago

0 replies

Concurrent lines

MathChallenger101 4

N an hour ago by oVlad

Let $A B C D$ be an inscribed quadrilateral. Circles of diameters $A B$ and $C D$ intersect at points $X_1$ and $Y_1$ , and circles of diameters $B C$ and $A D$ intersect at points $X_2$ and $Y_2$ . The circles of diameters $A C$ and $B D$ intersect in two points $X_3$ and $Y_3$ . Prove that the lines $X_1 Y_1, X_2 Y_2$ and $X_3 Y_3$ are concurrent.

4 replies

MathChallenger101

Feb 8, 2025

oVlad

an hour ago

Find the value

sqing 7

N an hour ago by giangtruong13

Source: Own

Let $a,b,c$ be distinct real numbers such that $\frac{a^2}{(a-b)^2}+ \frac{b^2}{(b-c)^2}+ \frac{c^2}{(c-a)^2} =1.$ Find the value of $\frac{a}{a-b}+ \frac{b}{b-c}+ \frac{c}{c-a}.$
Let $a,b,c$ be distinct real numbers such that $\frac{a^2}{(b-c)^2}+ \frac{b^2}{(c-a)^2}+ \frac{c^2}{(a-b)^2}=2.$ Find the value of $\frac{a}{b-c}+ \frac{b}{c-a}+ \frac{c}{a-b}.$
Let $a,b,c$ be distinct real numbers such that $\frac{(a+b)^2}{(a-b)^2}+ \frac{(b+c)^2}{(b-c)^2}+ \frac{(c+a)^2}{(c-a)^2}=2.$ Find the value of $\frac{a+b}{a-b}+\frac{b+c}{b-c}+ \frac{c+a}{c-a}.$

7 replies

sqing

Mar 17, 2025

giangtruong13

an hour ago

2025 HMIC-5

EthanWYX2009 0

an hour ago

Source: 2025 HMIC-5

Compute the smallest positive integer $k > 45$ for which there exists a sequence $a_1, a_2, a_3, \ldots ,a_{k-1}$ of positive integers satisfying the following conditions:[list]
[*] $a_i = i$ for all integers $1 \le i \le 45;$
[*] $a_{k-i} = i$ for all integers $1 \le i \le 45;$
[*] for any odd integer $1 \le n \le k -45,$ the sequence $a_n, a_{n+1}, \ldots , a_{n+44}$ is a permutation of
$\{1, 2, \ldots , 45\}.$ [/list]
Proposed by: Derek Liu

0 replies

EthanWYX2009

an hour ago

0 replies

I need the technique

DievilOnlyM 14

N an hour ago by Entei

Let a,b,c be real numbers such that: $ab+7bc+ca=188$ .
FInd the minimum value of: $5a^2+11b^2+5c^2$

14 replies

DievilOnlyM

May 23, 2019

Entei

an hour ago

Cyclic Quads and Parallel Lines

gracemoon124 15

N an hour ago by Adywastaken

Source: 2015 British Mathematical Olympiad?

Let $ABCD$ be a cyclic quadrilateral. Let $F$ be the midpoint of the arc $AB$ of its circumcircle which does not contain $C$ or $D$ . Let the lines $DF$ and $AC$ meet at $P$ and the lines $CF$ and $BD$ meet at $Q$ . Prove that the lines $PQ$ and $AB$ are parallel.

15 replies

gracemoon124

Aug 16, 2023

Adywastaken

an hour ago

Something nice

KhuongTrang 33

N an hour ago by NguyenVanHoa29

Source: own

Problem. Given $a,b,c$ be non-negative real numbers such that $ab+bc+ca=1.$ Prove that

$\sqrt{a+1}+\sqrt{b+1}+\sqrt{c+1}\le 1+2\sqrt{a+b+c+abc}.$

33 replies

KhuongTrang

Nov 1, 2023

NguyenVanHoa29

an hour ago

Geometry hard

Lukariman 2

N an hour ago by Primeniyazidayi

Given triangle ABC inscribed in circle (O). The bisector of angle A intersects (O) at D. Let M, N be the midpoints of AB, AC respectively. OD intersects BC at P and AD intersects MN at S. The circle circumscribed around triangle MPS intersects BC at Q different from P. Prove that QA is tangent to (O).

2 replies

Lukariman

2 hours ago

Primeniyazidayi

an hour ago

help!!!!!!!!!!!!

Cobedangiu 3

N 2 hours ago by sqing

help

$a,b,c\ge 0. a^2+b^2+c^2=8. Prove that: 4(a+b+c-4)\le abc

3 replies

Cobedangiu

Mar 23, 2025

sqing

2 hours ago

Inequality

nguyentlauv 1

N 2 hours ago by NguyenVanHoa29

Source: Own

Let $a,b,c$ be positive real numbers such that $ab+bc+ca=3$ and $k\ge 0$ , prove that
$\frac{\sqrt{a+1}}{b+c+k}+\frac{\sqrt{b+1}}{c+a+k}+\frac{\sqrt{c+1}}{a+b+k} \geq \frac{3\sqrt{2}}{k+2}.$

1 reply

1 viewing

nguyentlauv

Yesterday at 12:19 PM

NguyenVanHoa29

2 hours ago

Great similarity

steven_zhang123 3

N 2 hours ago by Lil_flip38

Source: a friend

As shown in the figure, there are two points $D$ and $E$ outside triangle $ABC$ such that $\angle DAB = \angle CAE$ and $\angle ABD + \angle ACE = 180^{\circ}$ . Connect $BE$ and $DC$ , which intersect at point $O$ . Let $AO$ intersect $BC$ at point $F$ . Prove that $\angle ACE = \angle AFC$ .

3 replies

steven_zhang123

2 hours ago

Lil_flip38

2 hours ago

Nice problem about the Lemoine point of triangle JaBC and OI line

Ktoan07 0

Apr 18, 2025

Source: Own

Let $\triangle ABC$ be an acute-angled, non-isosceles triangle with circumcenter $O$ and incenter $I$ , such that

$\prod_{\text{cyc}} \left( \frac{1}{a+b-c} + \frac{1}{a+c-b} - \frac{2}{b+c-a} \right) \neq 0,$
where $a = BC$ , $b = CA$ , and $c = AB$ .

Let $J_a$ , $J_b$ , and $J_c$ be the excenters opposite to vertices $A$ , $B$ , and $C$ , respectively, and let $L_a$ , $L_b$ , and $L_c$ be the Lemoine points of triangles $J_aBC$ , $J_bCA$ , and $J_cAB$ , respectively.

Prove that the circles $(L_aBC)$ , $(L_bCA)$ , and $(L_cAB)$ all pass through a common point $P$ . Moreover, the isogonal conjugate of $P$ with respect to $\triangle ABC$ lies on the line $OI$ .

Note (Hint)

There is a synethic solution, which use the common radical axis of three circles.

0 replies