Problem 02 (An Equilateral triangle)

10

D, E, F be midpoints of an equilateral triangle \Delta ABC. T be point lie on incircle of \Delta DEF.

Prove that \displaystyle BP+CQ+AR=\frac{3}{2}.AB

Author: Van Khea, Cambodia

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Problem 08

08

Line (d) tangent to incircle of an equilateral triangle \Delta ABC at P.

Prove that \displaystyle \frac{AP}{PD}+\frac{AQ}{QE}=1

Author: Van Khea, Cambodia

Line tangent to circumcircle

problem

\Delta ABC is an equilateral triangle. Prove that:

1) PD=PI+PE

2) \displaystyle PD^4+PE^4+PI^4=\frac{1}{2}.AB^4

Author: Van Khea, Cambodia

right triangle inequality

c1

Let \displaystyle \frac{AH}{HC}=\frac{7}{3}.

Prove that \displaystyle AB^3+BC^3<\frac{3}{4}.AC^3

Author: Van Khea, Cambodia

Sweet problem

NICE

Let O be circumcenter of triangle \Delta ABC.

Prove that:  \displaystyle \frac{DB\times DC}{AB\times AC}+\frac{EC\times EA}{BC\times BA}+\frac{FA\times FB}{CA\times CB}=1

Author: Van Khea, Cambodia

 

An Equilateral problems

p2

Let D, E, F be midpoints of an equilateral triangle \Delta ABC. Let K be midpoint of EF. Prove that:

a) \displaystyle PD^2+PE^2=\frac{32}{3}.PF^2

b) \displaystyle PD^4+PE^4=\frac{530}{9}.PF^4

Author: Van Khea, Cambodia

Geometry, triangle

p1

\Delta ABC is an equilateral triangle.

E, F is midpoints of CA, AB and CP\perp EF

Prove that \displaystyle sin^2(EFP)+sin^2(EPF)=\frac{1}{7}

Proposed by Van Khea, Cambodia