orthocentre and circumcentre

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0 replies

jlacosta

Apr 2, 2025

0 replies

a+b+c=abc

KhuongTrang 1

N 9 minutes ago by KhuongTrang

Source: own

Problem. Let $a,b,c$ be three positive real numbers satisfying $a+b+c=abc.$ Prove that $\sqrt{a^2+b^2+3}+\sqrt{b^2+c^2+3}+\sqrt{c^2+a^2+3}\ge4\cdot \frac{a^2b^2c^2-3}{ab+bc+ca-3}-7.$ There is a very elegant proof :-D Could anyone think of it?

1 reply

1 viewing

KhuongTrang

Wednesday at 11:51 AM

KhuongTrang

9 minutes ago

multiple of 15-15 positive factors

britishprobe17 0

14 minutes ago

Source: KTOM Maret 2025

Find the sum of all natural numbers $n$ such that $n$ is a multiple of $15$ and has exactly $15$ positive factors.

0 replies

britishprobe17

14 minutes ago

0 replies

general form

pennypc123456789 0

19 minutes ago

If $a,b,c$ are positive real numbers, $k \ge 3$ then
$\frac{a + b}{a + kb + c} + \dfrac{b + c}{b + kc + a}+\dfrac{c + a}{c + ka + b} \geq \dfrac{6}{k+2}$

0 replies

pennypc123456789

19 minutes ago

0 replies

Function equation

luci1337 2

N 26 minutes ago by pco

find all function $f:R \rightarrow R$ such that:
$2f(x)f(x+y)-f(x^2)=\frac{x}{2}(f(2x)+f(f(y)))$ with all $x,y$ is real number

2 replies

luci1337

Yesterday at 3:01 PM

pco

26 minutes ago

No more topics!

orthocentre and circumcentre

AndrewTom 1

N Dec 11, 2012 by Lyub4o

Vertex $A$ of triangle $ABC$ is equidistant from the circumcentre $O$ and the orthocentre $H$ . Find all possible values of angle $A$ .

1 reply

AndrewTom

Dec 11, 2012

Lyub4o

Dec 11, 2012

orthocentre and circumcentre

G H J

Reveal topic tags

geometry unsolved

geometry

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AndrewTom

12750 posts

AndrewTom

#1 h Dec 11, 2012, 7:41 PM • 2 Y

Y by Adventure10, Mango247

Vertex $A$ of triangle $ABC$ is equidistant from the circumcentre $O$ and the orthocentre $H$ . Find all possible values of angle $A$ .

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Lyub4o

265 posts

Lyub4o

#2 h Dec 11, 2012, 7:54 PM • 3 Y

Y by AndrewTom, Adventure10, Mango247

Pretty problem.Let $O$ be inside triangle $ABC$ .Let $M$ be the middle of $BC$ .It is famous fact that $2.OM=AH$ .From there in triangle $MBO$ we have that $BO=2OM$ ,which for right triangle means $\angle OBM=30$ .So $\angle BOC=120$ and $\angle A=60$ .If $O$ is outside triangle $ABC$ we get $\angle A=120$ in a similar manner.

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