Given: a + b + 2c = 1.

Find the minimum and maximum of A = ab - 4bc - 4ca

5. Given the triangle ABC balance at A (AB = AC). On BC take M any such that BM < CM. From M draw parallel lines with AC cut AB at E and parallel with AB cut AC at F. Let N be the symmetry point of M over EF.
a) Calculate the perimeter of AEMF. Know: AB = 7cm
b) Prove that: AFEN is a trapezoid trait
c) Calculated: \(\widehat{ANB}+\widehat{ACB}\)
d) Where M is the AEMF quadrilateral is diamond and needs to be added to the triangle ABC so that the AEMF is square.

4. (easy) Via the center of the triangle G, the straight line parallel to AC, cut AB and BC respectively at M and N. Calculate the length of the field, know AM + NC = 16 (cm); Triangle triangle ABC equals 75 cm

3. Given triangle ABC. On BC, CA, AB take D, E, F so that AD, BE, DF intersect at H.

Prove that:

a, \(\dfrac{AH}{AD}+\dfrac{BH}{BE}+\dfrac{CH}{CF}=2\)

b, \(\dfrac{AH}{HD}+\dfrac{BH}{HE}+\dfrac{CH}{HF}\ge6\)

2. Given triangle ABC. M is a point in the interior of triangle ABC. Let D, E, F be the midpoints AB, AC, BC; A ', B', C 'is the symmetry point of M through F, E, D.
a, CMR: AB'A'B is a parallelogram.
b, CMR: CC 'goes through the midpoint of AA'

In this web - Mathulike mostly is algebraic and only few geometry. So I will post some exercises of geometry.

1. Given ABCD rectangle with AB = 2AD, call E, I in turn is the midpoint of AB and CD. Connect D to E. Draw the Dx beam perpendicular to DE, the Dx beam to the opposite ray of the CB ray at M. On the opposite side of the CE beam take the K point so that DM = EK. Let G be the intersection of DK and EM.
a) Calculate DBK angle.
b) Let F be the right-angled foot from K to BM. Prove that four points A, I, G, H are in the same line.

In recent times, I have seen some questions not related to mathematics. And that is not good. I suspect there are some accounts where the main account is https://mathulike.com/profile/ducanh2007 has posted the question itself and tick yourself. As an coodinator, I hope Admin will soon resolve this issue. And I advise you to have the nickname "Cristiano Ronaldo" stop ticking yourself. I hope Mr.Bee will solve it satisfactorily.

And to stop this status, I will post some question in my free time. 

With all a,b.

Prove that: a, \(\left(a+b\right)^2\ge4ab\)

                   b, \(a+b\ge2\sqrt{ab}\)

Simple 

With all a,b

Prove that: \(a^4+b^4\ge a^3b+b^3a\)

Given: \(a,b\in Z\)

Prove that: \(\left(a+b+c\right)^3\ge3\left(ab+bc+ca\right)\)

Given: a,b > 0

Prove that: \(\dfrac{a^3+b^3}{2}\ge\left(\dfrac{a+b}{2}\right)^3\)

Given: x + y = 2 

Prove that: \(a^4+b^4\ge2\)

Given: a + b > 1

Prove that: \(a^4+b^4>\dfrac{1}{8}\)

A B C D E F

Prove that:

\(\dfrac{3}{4}\left(AB+AC+BC\right)< AD+BE+CF< AB+AC+BC\)

Caculator: 

\(A=\dfrac{\dfrac{\dfrac{2017}{1}+\dfrac{2016}{2}+...+\dfrac{1}{2017}}{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2018}}}{\dfrac{\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2017}}{\dfrac{1}{2016}+\dfrac{2}{2015}+...+\dfrac{2016}{1}}}\)

Given: (2x₁ - 5x₁)²⁰¹⁸ + (2x₂ - 5x₂)²⁰¹⁸ + ... + (2x₂₀₁₉ - 5x₂₀₁₉)²⁰¹⁸ \(\le\)0

Prove that: \(\dfrac{x_1+x_2+...+x_{2019}}{y_1+y_2+...+y_{2019}}=2,5\)

Find the minimum of P = |7x - 5y| + |2z - 3x| + |xy + yz + zx - 2000| 

Calculator:

\(\left(1-\dfrac{1}{1+2}\right)\left(1-\dfrac{1}{1+2+3}\right)\left(1-\dfrac{1}{1+2+3+4}\right)...\left(1-\dfrac{1}{1+2+3+...+2006}\right)\)

Question 2: Prove that if a,b,c and \(\sqrt{a}+\sqrt{b}+\sqrt{c}\) are rational numbers then \(\sqrt{a},\sqrt{b},\sqrt{c}\) are rational numbers

Today and next days, I will post some questions everyday. 

Question 1: Find the integer part of \(\sqrt{2}+\sqrt[3]{\dfrac{3}{2}}+\sqrt[4]{\dfrac{4}{3}}+...+\sqrt[n+1]{\dfrac{n+1}{n}}\)