### Several inequalities of Erdős-Mordell type

Mihály Bencze and Marius Drăgan

Str. Hărmanului 6, 505600 Săcele-Négyfalu, Jud. Braşov, Romania E-mail: [email protected]

61311 bd. Timişoara Nr. 35, Bl. 0D6, Sc. E, et. 7, Ap. 176, Sect. 6, Bucureşti, Romania E-mail: [email protected]

ABSTRACT. The purpose of this paper is to obtain some inequalities between sum or product of power to
order,*k∈*[0,1]relate to distences from an interior point*M* of triangle*ABC* to the vertices of triangle, radii of
triangle*M BC, M CA, M AB*and distances from M to the sides*BC, CA, AB* and using this inequalities and an
inversion of pol*M* and ratio*t*in conexion with vertices*A, B, C*to obtain many other new inequalities.

1 INTRODUCTION

Let be M an interior point of triangle ABC,*R*_{a}*, R*_{b}*, R** _{c}* the radii of the circumscribe of triangles

*M BC, M CA, M AB, R*_{1}*, R*_{2}*, R*_{3}the distances from*M* to the vertices*A, B, C* and*r*_{1}*, r*_{2}*, r*_{3} the distances
from*M* to the sides*BC, CA, AB.*

The inequality Erdős-Mordell

*R*_{1}+*R*_{2}+*R*_{3}*≥*2 (r_{1}+*r*_{2}+*r*_{3}) (1.1)
is true in any triangle ABC.

Also a generalization of Erdős-Mordell appears in [1] who said that:

*x*^{2}*R*1+*y*^{2}*R*2+*z*^{2}*R*3*≥*2 (yzr1+*zxr*2+*xyr*3) (1.2)
is true for any real numbers*x, y, z≥*0.

If we take*λ*1=*x*^{2}*, λ*2=*y*^{2}*, λ*3=*z*^{2} are true inequality
*λ*1*R*1+*λ*2*R*2+*λ*3*R*3*≥*2(√

*λ*2*λ*3*r*1+√

*λ*3*λ*1*r*2+√
*λ*1*λ*2*r*3

)

(1.3)

who appears in [2].

Also

*λ*^{2}_{1}*R**a*+*λ*^{2}_{2}*R**b*+*λ*^{2}_{3}*R**c**≥λ*1*λ*2*λ*3

(*R*1

*λ*_{1} +*R*2

*λ*_{2} +*R*3

*λ*_{3}
)

(1.4)

Who are published in [3].

In the following we give a lot of inequalities of this type and many other in connection with them.

Lemma 1. Let be*M* an interior point of triangle *ABC.*Then we have:

a). 1*≥*_{a}^{c}*·*_{2R}^{R}^{3}* _{b}* +

^{b}

_{a}*·*

_{2R}

^{R}^{2}

*b). 1*

_{c}*≥*

_{a}

^{b}*·*

_{2R}

^{R}^{3}

*+*

_{b}

_{a}

^{c}*·*

_{2R}

^{R}^{2}

_{c}*Proof.*

a). We use the well-known inequality

*aR*1*≥cr*2+*br*3

and the equality

*r*1= *R*2*R*3

2R* _{a}*
who result writing the area of triangle MBC

*S*_{△}*ABC* =*R*2*R*3

4R* _{a}* =

*ar*1

2 b). It result from the inequality

*aR*1*≥br*2+*cr*3

Lemma 2. Let be*M* an interior point of triangle *ABC.*Then we have:

*R*_{2}
*R**c*

+*R*_{3}

*R**b* *≤*4 sin*A*
2

We consider*U* =*pr*_{AB}^{M}*, V* =*pr*^{M}_{AC}*. u*=*µ*(∡*U AM), v*=*µ*(∡*V AM).*We have
*r*_{2}+*r*_{3}=*R*_{1}(sin*u*+ sin*v) = 2R*_{1}sin*A*

2 cos*u−v*

2 *≤*2R_{1}sin*A*
2
or

*R*3*R*1

2R* _{b}* +

*R*1

*R*2

2R_{c}*≤*2R1sin*A*
2
Who is equivalent with the inequality from the statement.

Theorem 1. Let be*M* a interior point of triangle *ABC. Then for every real numberk∈*[0,1],
*λ*_{1}*, λ*_{2}*, λ*_{3}*≥*0.We have:

a). *λ*_{1}*R*^{k}* _{a}*+

*λ*

_{2}

*R*

^{k}*+*

_{b}*λ*

_{3}

*R*

^{k}

_{c}*≥√*

*λ*_{2}*λ*_{3}*R*^{k}_{1}+*√*

*λ*_{3}*λ*_{1}*R*_{2}* ^{k}*+

*√*

*λ*

_{1}

*λ*

_{2}

*R*

^{k}_{3}b).

*√*

*λ*_{2}*λ*_{3}_{R}_{k}^{R}^{k}^{1}

*b*+R^{k}* _{c}* +

*√*

*λ*_{3}*λ*_{1}_{R}_{k}^{R}^{k}^{2}

*c*+R_{a}* ^{k}*+

*√*

*λ*_{1}*λ*_{2}_{R}_{k}^{R}^{k}^{3}

*a*+R^{k}_{b}*≤*

*≤* ^{1}_{2}
(*√*

*λ*2*λ*3
*R*^{k}_{1}

*√**R*^{k}_{b}*R*^{k}* _{c}* +

*√*

*λ*3

*λ*1

*R*^{k}_{2}

*√**R*^{k}_{c}*R*^{k}* _{a}* +

*√*

*λ*1

*λ*2

*R*^{k}_{3}

*√**R*^{k}_{a}*R*^{k}_{b}

)

*≤* ^{λ}^{1}^{+λ}_{2}^{2}^{+λ}^{3}
*Proof.* a). From the inequality(x+*y)*^{k}*≥*2^{k}^{−}^{1}(

*x** ^{k}*+

*y*

*)*

^{k}*, k∈*[0,1]it result that:

(*a*
*c* *·* *R*2

2R* _{a}* +

*b*

*c*

*·*

*R*1

2R* _{b}*
)

*k*

*≥*2^{k}^{−}^{1}
([(*a*

*c*

)*k* *R*^{k}_{2}
(2R*a*)* ^{k}* +

(*b*
*c*

)*k*(
*R*1

2R* _{b}*
)

*k*])

Using this inequality and Lemma 1. We obtain

1*≥*
(*a*

*c* *·* *R*2

2R* _{a}* +

*b*

*c·*

*R*1

2R* _{b}*
)

*k*

*≥*2^{k}^{−}^{1}
[(*a*

*c*

)*k* *R*^{k}_{2}
(2R* _{a}*)

*+*

^{k}(*b*
*c*

)*k*(
*R*^{k}_{1}
2R_{b}

)*k*]

or

2*≥a*^{k}*c*^{k}*·R*_{2}^{k}

*R*_{a}* ^{k}* +

*b*

^{k}*c*

^{k}*·R*

^{k}_{1}

*R*^{k}* _{b}*
or

2λ3*R*^{k}_{c}*≥λ*3

*a*^{k}*c*^{k}*·* *R*^{k}_{c}

*R*^{k}_{a}*·R*^{k}_{2}+*λ*3

*b*^{k}*c*^{k}*·R*^{k}_{c}

*R*^{k}_{b}*·R*^{k}_{1}
and the similar inequalities

2λ_{2}*R*^{k}_{b}*≥λ*_{2}*c*^{k}*b*^{k}*·R*^{k}_{b}

*R*^{k}_{c}*·R*^{k}_{1}+*λ*_{2}*a*^{k}*b*^{k}*·R*^{k}_{b}

*R*^{k}_{a}*·R*^{k}_{3}

2λ_{1}*R*^{k}_{a}*≥λ*_{1}*b*^{k}*s*^{k}*·R*^{k}_{a}

*R*^{k}_{b}*·R*^{k}_{3}+*λ*_{1}*c*^{k}*a*^{k}*·* *R*^{k}_{a}

*R*^{k}_{c}*·R*^{k}_{2}
Adding this inequalities and apply the A-M inequalities we obtain:

2(

*λ*_{1}*R*^{k}* _{a}*+

*λ*

_{2}

*R*

_{b}*+*

^{k}*λ*

_{3}

*R*

^{k}*)*

_{c}*≥R*^{k}_{1}
(

*λ*_{3}*·b*^{k}*c*^{k}*·* *R*^{k}_{c}

*R*^{k}* _{b}* +

*λ*

_{2}

*·*

*c*

^{k}*b*

^{k}*·*

*R*

^{k}

_{b}*R*^{k}* _{c}*
)

+

+R_{2}* ^{k}*
(

*λ*_{3}*·a*^{k}*c*^{k}*·R*^{k}_{c}

*R*^{k}* _{a}* +

*λ*

_{1}

*·*

*c*

^{k}*a*

^{k}*·*

*R*

^{k}

_{a}*R*^{k}* _{c}*
)

+*R*^{k}_{3}
(

*λ*_{2}*·* *a*^{k}*b*^{k}*·R*^{k}_{b}

*R*^{k}* _{a}* +

*λ*

_{1}

*·*

*b*

^{k}*a*

^{k}*·R*

^{k}

_{a}*R*^{k}* _{b}*
)

*≥*

*≥*2√

*λ*2*λ*3*R*_{1}* ^{k}*+ 2√

*λ*3*λ*1*R*^{k}_{2}+ 2√
*λ*2*λ*1*R*^{k}_{3}

b). Adding the inequalities

2λ3*≥λ*3*·a*^{k}*c*^{k}*·* *R*^{k}_{2}

*R*^{k}* _{a}* +

*λ*3

*·*

*b*

^{k}*c*

^{k}*·R*

^{k}_{1}

*R*^{k}_{b}

2λ1*≥λ*1*·* *b*^{k}*a*^{k}*·R*^{k}_{3}

*R*^{k}* _{b}* +

*λ*1

*·*

*c*

^{k}*a*

^{k}*·*

*R*

^{k}_{2}

*R*^{k}_{c}

2λ3*≥λ*2*·c*^{k}*b*^{k}*·R*^{k}_{1}

*Rc*+*λ*2*·a*^{k}*b*^{k}*·R*^{k}_{3}

*R*^{k}* _{a}*
We obtain

2 (λ_{1}+*λ*_{2}+*λ*_{3})*≥*
(

*λ*_{3}*·a*^{k}*c*^{k}*·* *R*^{k}_{2}

*R*^{k}* _{a}* +

*λ*

_{1}

*·*

*c*

^{k}*a*

^{k}*·R*

^{k}_{2}

*R*^{k}* _{c}*
)

+

+ (

*λ*_{3}*·b*^{k}*c*^{k}*·R*^{k}_{1}

*R*^{k}* _{b}* +

*λ*

_{2}

*·c*

^{k}*b*

^{k}*·R*

^{k}_{1}

*R*^{k}* _{c}*
)

+ (

*λ*_{1}*·* *b*^{k}*a*^{k}*·R*^{k}_{3}

*R*^{k}* _{b}* +

*λ*

_{2}

*·*

*a*

^{k}*b*

^{k}*·R*

^{k}_{3}

*R*^{k}* _{a}*
)

*≥*

*≥*2√

*λ*1*λ*3*·* *R*^{k}_{2}

√*R*^{k}_{a}*R*^{k}* _{c}* + 2√

*λ*2*λ*3*·* *R*^{k}_{1}

√
*R*^{k}_{b}*R*^{k}_{c}

+ 2√

*λ*1*λ*2*·* *R*^{k}_{3}

√
*R*^{k}_{a}*R*^{k}_{b}

*≥*

*≥*4√

*λ*_{1}*λ*_{3}*·* *R*^{k}_{2}

*R*^{k}* _{a}*+

*R*

^{k}*+ 4√*

_{c}*λ*_{2}*λ*_{3}*·* *R*^{k}_{1}

*R*^{k}* _{b}*+

*R*

^{k}*+ 4√*

_{c}*λ*_{1}*λ*_{2}*·* *R*^{k}_{3}
*R*_{a}* ^{k}*+

*R*

^{k}

_{b}We observe that b) ^{imply}*⇒* a)

Indeed putting in b). *λ*1*→λ*1*R*^{k}_{a}*, λ*2*→λ*2*R*^{k}_{b}*, λ*3*→λ*3*R*^{k}_{c}*,*we obtain

√

*R*^{k}_{b}*R*_{c}^{k}*·* *R*^{k}_{1}*·√*
*λ*2*λ*3

*R*_{b}* ^{k}*+

*R*

_{c}*+*

^{k}√

*R*^{k}_{c}*R*^{k}_{a}*·* *R*^{k}_{2}*·√*
*λ*3*λ*1

*R*^{k}* _{a}*+

*R*

^{k}*+*

_{c}√

*R*^{k}_{a}*R*^{k}_{b}*·R*^{k}_{3}*·√*
*λ*1*λ*2

√

*R*^{k}* _{a}*+

*R*

^{k}

_{b}*≤*

1 2

(*λ*_{1}*R*^{k}_{1}+*λ*_{2}*R*^{k}_{2}+*λ*_{3}*R*^{k}_{3})

*≤*1
2

(*λ*_{1}*R*_{a}* ^{k}*+

*λ*

_{2}

*R*

^{k}*+*

_{b}*λ*

_{3}

*R*

_{c}*)*

^{k}Also putting in b). *λ*_{1}*→λ*_{1}*R*^{k}_{b}*, λ*_{2}*→λ*_{2}*R*^{k}_{c}*, λ*_{3}*→λ*_{3}*R*^{k}* _{a}* and

*λ*

_{1}

*→λ*

_{2}

*R*

^{k}

_{b}*, λ*

_{2}

*→λ*

_{3}

*R*

^{k}

_{c}*, λ*

_{3}

*→λ*

_{1}

*R*

^{k}

_{a}*,*we obtain:

Theorem 2. Let be*M* an interior point of triangle*ABC. Then for ecery real numberk∈*[0,1],
*λ*_{1}*, λ*_{2}*, λ*_{3}*≥*0, we have:

a).

√*λ*_{2}*λ*_{3}*·R*^{k}_{1}√
*R*^{k}_{c}*·R*^{k}_{a}*R*^{k}* _{b}*+

*R*

^{k}*+√*

_{c}*λ*_{3}*λ*_{1}*·* *R*^{k}_{2}

√
*R*^{k}_{a}*·R*^{k}_{b}*R*^{k}* _{a}*+

*R*

^{k}*+√*

_{c}*λ*_{1}*λ*_{2}*·R*^{k}_{3}

√
*R*^{k}_{b}*·R*^{k}_{c}*R*^{k}* _{a}*+

*R*

^{k}

_{b}*≤*

*≤*1
2

√
*λ*2*λ*3*R*^{k}_{1}

√
*R*^{k}_{a}*R*^{k}* _{b}* +√

*λ*3*λ*1*R*^{k}_{2}

√
*R*^{k}_{b}*R*^{k}* _{c}* +√

*λ*1*λ*2*R*_{3}^{k}

√
*R*^{k}_{c}*R*^{k}_{a}

*≤*

*≤* 1
2

(*λ*1*R*^{k}* _{b}* +

*λ*2

*R*

^{k}*+*

_{c}*λ*3

*R*

^{k}*) b).*

_{a}√*λ*3*λ*1*·R*_{1}* ^{k}*√

*R*

^{k}

_{c}*·R*

^{k}

_{a}*R*

^{k}*+*

_{b}*R*

^{k}*+√*

_{c}*λ*1*λ*2*·* *R*^{k}_{2}

√
*R*^{k}_{a}*·R*_{b}^{k}*R*^{k}* _{c}* +

*R*

^{k}*+√*

_{a}*λ*2*λ*3*·R*^{k}_{3}

√
*R*_{b}^{k}*·R*^{k}_{c}*R*^{k}* _{a}*+

*R*

^{k}

_{b}*≤*

*≤*1
2

√
*λ*_{3}*λ*_{1}*R*^{k}_{1}

√
*R*^{k}_{a}*R*^{k}* _{b}* +√

*λ*_{1}*λ*_{2}*R*^{k}_{2}

√
*R*^{k}_{b}*R*^{k}* _{c}* +√

*λ*_{2}*λ*_{3}*R*^{k}_{3}

√
*R*^{k}_{c}*R*^{k}_{a}

*≤*

*≤* 1
2

(*λ*_{2}*R*^{k}* _{b}* +

*λ*

_{3}

*R*

^{k}*+*

_{c}*λ*

_{1}

*R*

^{k}*)*

_{a}If in Theorem 1 and 2 we take*λ*_{1}=*λ*_{2}=*λ*_{3} we obtain

Corollary 2.1. a).

*R*^{k}* _{a}*+

*R*

^{k}*+*

_{b}*R*

^{k}

_{c}*≥R*

^{k}_{1}+

*R*

^{k}_{2}+

*R*

^{k}_{3}b).

*R*^{k}_{1}

*R*^{k}* _{b}* +

*R*

^{k}*+*

_{c}*R*

^{k}_{2}

*R*^{k}* _{c}* +

*R*

^{k}*+*

_{a}*R*

^{k}_{3}

*R*^{k}* _{a}*+

*R*

^{k}

_{b}*≤*1 2

*R*^{k}_{1}

√
*R*^{k}_{b}*R*^{k}_{c}

+ *R*^{k}_{2}

√*R*^{k}_{c}*R*^{k}* _{a}* +

*R*

_{3}

^{k}√
*R*^{k}_{a}*R*^{k}_{b}

*≤*3
2

c).

*R*^{k}_{1}√
*R*^{k}_{c}*R*^{k}_{a}*R*^{k}* _{b}* +

*R*

_{a}*+*

^{k}*R*^{k}_{2}

√
*R*^{k}_{a}*R*_{b}^{k}*R*^{k}* _{a}*+

*R*

^{k}*+*

_{c}*R*^{k}_{3}

√
*R*^{k}_{b}*R*^{k}_{c}*R*^{k}* _{a}*+

*R*

^{k}

_{b}*≤*1

2

R^{k}_{1}

√
*R*^{k}_{a}*R*^{k}* _{b}* +

*R*

^{k}_{2}

√
*R*^{k}_{b}*R*^{k}* _{c}* +

*R*

^{k}_{3}

√
*R*^{k}_{c}*R*^{k}_{a}

*≤*

*≤* 1
2

(*R*^{k}* _{a}*+

*R*

^{k}*+*

_{b}*R*

^{k}*)*

_{c}If we take *k*= 1in a). we obtain the following inequality

*R**a*+*R**b*+*R**c**≥R*1+*R*2+*R*3

who appears in [1].

Also if we take*k*= 1 in b). we obtain:

*R*1

*R** _{b}*+

*R*

*+*

_{c}*R*2

*R** _{c}*+

*R*

*+*

_{a}*R*3

*R** _{a}*+

*R*

_{b}*≤*1 2

( *R*1

*√R**b**R**c*

+ *R*2

*√R**a**R**b*

+ *R*3

*√R**a**R**b*

)

who represent a refinement of inequality _{R}^{R}^{1}

*b*+R* _{c}* +

_{R}

^{R}^{2}

*c*+R* _{a}* +

_{R}

^{R}^{3}

*a*+R_{b}*≤*^{3}_{2} who apperas in [2].

If we take in c). *k*= 1we obtain
*R*1

*√R**c**R**a*

*R** _{b}*+

*R*

*+*

_{c}*R*2

*√R**a**R**b*

*R** _{a}*+

*R*

*+*

_{c}*R*3

*√R**b**R**c*

*R** _{a}*+

*R*

_{c}*≤*1 2

(
*R*1

√*R**a*

*R** _{b}* +

*R*2

√*R**b*

*R** _{c}* +

*R*3

√*R**c*

*R** _{a}*
)

*≤*

*≤* 1

2(R*a*+*R**b*+*R**c*)
If in Theorem 2) we take*λ*1*→λ*^{2}_{1}*, λ*2*→λ*^{2}_{2}*, λ*3*→λ*^{2}_{3}*,*we obtain

Corollary 2.2. Let be*M* an interior point of triangle*ABC. Then for every real numberk∈*[0,1],
*λ*1*, λ*2*, λ*3*≥*0, we have

*λ*^{2}_{1}*R*^{k}* _{a}*+

*λ*

^{2}

_{2}

*R*

^{k}*+*

_{b}*λ*

^{2}

_{3}

*R*

^{2}

_{c}*≥λ*1

*λ*2

*λ*3

(*R*^{k}_{1}
*λ*_{1} +*R*^{k}_{2}

*λ*_{2} +*R*^{k}_{3}
*λ*_{3}

)
If we take *k*= 1we obtain

*λ*^{2}_{1}*R**a*+*λ*^{2}_{2}*R**b*+*λ*^{2}_{3}*R**c**≥λ*1*λ*2*λ*3

(*R*1

*λ*_{1} +*R*2

*λ*_{2} +*R*3

*λ*_{3}
)

who represent just inequality (1.4).

Corollary 2.3. Let be M an interior point of triangle*ABC.*Then for every real number*k∈*[0,1]we
have

a).

*R*^{k}_{1}*R*^{k}* _{a}*+

*R*

^{k}_{2}

*R*

^{k}*+*

_{b}*R*

^{k}_{3}

*R*

^{k}

_{c}*≤R*

^{k}

_{a}*R*

_{b}*+*

^{k}*R*

^{k}

_{b}*R*

^{k}*+*

_{c}*R*

_{c}

^{k}*R*

^{k}*b).*

_{a}1 + *R*^{k}_{b}*R*^{k}* _{c}* +

*R*

^{k}

_{c}*R*^{k}_{b}*≥* *R*^{k}_{1}

√
*R*^{k}_{b}*R*^{k}_{c}

+ *R*^{k}_{2}

√
*R*^{k}_{a}*R*^{k}_{b}

+ *R*^{k}_{3}

√*R*^{k}_{a}*R*^{k}_{c}

and similar inequalities c).

*R*^{k}_{1}*R*^{k}* _{a}*+

*R*

_{2}

^{k}√

*R*^{k}_{b}*R*^{k}* _{c}* +

*R*

^{k}_{3}

√

*R*^{k}_{b}*R*^{k}_{c}*≤R*^{k}_{a}*R*^{k}* _{b}* +

*R*

^{k}

_{b}*R*

^{k}*+*

_{c}*R*

^{k}

_{c}*R*

^{k}*and similar inequalities*

_{b}d).

*R*^{k}_{a}*R*^{k}* _{b}* +

*R*

^{k}

_{b}*R*^{k}* _{c}* +

*R*

_{c}

^{k}*R*_{a}^{k}*≥* *R*^{k}_{1}

√*R*^{k}_{c}*R*^{k}* _{a}* +

*R*

^{k}_{2}

√
*R*^{k}_{b}*R*^{k}_{a}

+ *R*^{k}_{3}

√
*R*^{k}_{b}*R*_{c}^{k}

and similar inequalities e).

*R*^{k}_{1}

√

*R*_{a}^{k}*R*^{k}* _{b}* +

*R*

^{k}_{2}

√

*R*^{k}_{b}*R*^{k}* _{c}* +

*R*

^{k}_{3}

√

*R*^{k}_{a}*R*^{k}_{c}*≤R*^{k}_{a}*R*^{k}* _{b}* +

*R*

^{k}

_{b}*R*

^{k}*+*

_{c}*R*

^{k}

_{c}*R*

^{k}

_{b}*Proof.*We take in theorem 1

(λ_{1}*, λ*_{2}*, λ*_{3}) =
( 1

*R*_{a}^{k}*,* 1
*R*_{b}^{k}*,* 1

*R*_{c}* ^{k}*
)

*,*(λ_{1}*, λ*_{2}*, λ*_{3}) =
( 1

*R*^{k}_{a}*,* 1
*R*^{k}_{b}*,* 1

*R*^{k}* _{c}*
)

(λ1*, λ*2*, λ*3) =
( 1

*R*_{b}^{k}*,* 1
*R*_{a}^{k}*,* 1

*R*_{c}* ^{k}*
)

*,*(λ1*, λ*2*, λ*3) =
( 1

*R*^{k}_{b}*,* 1
*R*^{k}_{c}*,* 1

*R*^{k}* _{a}*
)

(λ_{1}*, λ*_{2}*, λ*_{3}) =
( 1

*R*_{c}^{k}*,* 1
*R*_{a}^{k}*,* 1

*R*_{b}* ^{k}*
)

*,*(λ_{1}*, λ*_{2}*, λ*_{3}) =
( 1

*R*^{k}_{c}*,* 1
*R*^{k}_{b}*,* 1

*R*^{k}* _{a}*
)

Corollary 2.4. Let be M an interior point of triangle*ABC.*From every real number *k∈*[0,1]*,*we
have:

a).

*R*^{k}_{1}
*R*^{k}* _{b}* +

*R*

_{2}

^{k}*R*_{c}* ^{k}* +

*R*

^{k}_{3}

*R*

^{k}

_{a}*≤R*

_{b}

^{k}*R*_{c}* ^{k}* +

*R*

^{k}

_{c}*R*

^{k}*+*

_{a}*R*

^{k}

_{a}*R*^{k}_{b}

b).

*R*^{k}_{1}
*R*^{k}_{b}

√
*R*^{k}_{a}*R*^{k}* _{c}* +

*R*

^{k}_{2}

*R*^{k}_{c}

√
*R*^{k}_{b}*R*^{k}* _{a}* +

*R*

^{k}_{3}

*R*^{k}_{a}

√
*R*^{k}_{c}*R*^{k}_{b}*≤R*_{b}^{k}

*R*_{a}* ^{k}* +

*R*

^{k}

_{c}*R*

^{k}*+*

_{b}*R*

^{k}

_{a}*R*^{k}_{c}

c).

*R*^{k}_{1}
*R*^{k}* _{b}* +

*R*

_{2}

^{k}*R*_{c}^{k}

√
*R*^{k}_{b}

*R*^{k}* _{a}* +

*R*

^{k}_{3}

√
*R*^{k}_{b}*R*^{k}_{a}

*≤R*^{k}_{b}*R*^{k}* _{c}* +

*R*

^{k}

_{c}*R*^{k}* _{b}* + 1

and similar inequalities d).

*R*^{k}_{1}

√
*R*^{k}_{b}*R*^{k}_{c}

+ *R*^{k}_{2}

√*R*^{k}_{c}*R*^{k}* _{a}* +

*R*

^{k}_{3}

√
*R*^{k}_{a}*R*^{k}_{b}

*≤*3

*Proof.* We take in Theorem 2:

(λ_{1}*, λ*_{2}*, λ*_{3}) =
( 1

*R*_{a}^{k}*,* 1
*R*^{k}_{b}*,* 1

*R*^{k}* _{c}*
)

*,*(λ_{1}*, λ*_{2}*, λ*_{3}) =
( 1

*R*^{k}_{a}*,* 1
*R*^{k}_{b}*,* 1

*R*^{k}* _{c}*
)

(λ1*, λ*2*, λ*3) =
( 1

*R*_{b}^{k}*,* 1
*R*^{k}_{a}*,* 1

*R*^{k}* _{c}*
)

*,*(λ1*, λ*2*, λ*3) =
( 1

*R*^{k}_{b}*,* 1
*R*^{k}_{c}*,* 1

*R*^{k}* _{a}*
)

(λ1*, λ*2*, λ*3) =
( 1

*R*_{c}^{k}*,* 1
*R*^{k}_{a}*,* 1

*R*^{k}* _{b}*
)

*,*(λ1*, λ*2*, λ*3) =
( 1

*R*^{k}_{c}*,* 1
*R*^{k}_{b}*,* 1

*R*^{k}* _{a}*
)

Corollary 2.5. Let be M an interior point of triangle*ABC.* Then for every real number*k∈*[0,1]we
have

a).

*R*_{1}^{2k}*R*^{k}* _{c}* +

*R*

^{2k}

_{2}

*R*

^{k}*+*

_{a}*R*

^{2k}

_{3}

*R*

^{k}

_{b}*≥R*

^{2k}

_{1}

*R*

_{3}

*+*

^{k}*R*

^{2k}

_{2}

*R*

^{k}_{1}+

*R*

_{3}

^{2k}

*R*

^{k}_{2}b).

*R*^{2k}_{1} *R*^{k}_{3}

√
*R*^{k}_{b}*R*_{c}^{k}

+ *R*^{2k}_{2} *R*^{k}_{1}

√*R*^{k}_{c}*R*^{k}* _{a}* +

*R*

^{2k}

_{3}

*R*

^{k}_{2}

√
*R*_{a}^{k}*R*^{k}_{b}

*≤R*^{2k}_{1} +*R*^{2k}_{2} +*R*^{2k}_{3}

c).

*R*_{1}^{2k}*R*^{k}_{3}

√
*R*^{k}_{a}

*R*^{k}* _{b}* +

*R*

^{2k}

_{2}

*R*

_{1}

^{k}√
*R*^{k}_{b}

*R*^{k}* _{c}* +

*R*

^{2k}

_{3}

*R*

^{k}_{2}

√
*R*^{k}_{c}

*R*^{k}_{a}*≤R*^{2k}_{2} *R*_{b}* ^{k}*+

*R*

^{2k}

_{3}

*R*

^{k}*+*

_{c}*R*

_{1}

^{2k}

*R*

^{k}*d).*

_{a}*R*^{2k}_{1} *R*^{k}_{2}

√
*R*^{k}_{a}

*R*^{k}* _{b}* +

*R*

^{2k}

_{2}

*R*

^{k}_{3}

√
*R*^{k}_{b}

*R*^{k}* _{c}* +

*R*

^{2k}

_{3}

*R*

_{1}

√
*R*^{k}_{c}

*R*^{k}_{a}*≤R*^{2k}_{3} *R*^{k}* _{b}* +

*R*

^{2k}

_{1}

*R*

^{k}*+*

_{c}*R*

^{2k}

_{2}

*R*

^{k}

_{a}*Proof.*We take

(λ1*, λ*2*, λ*3) =
( *R*^{k}_{2}

*R*^{k}_{1}*R*^{k}_{3}*,* *R*^{k}_{3}
*R*^{k}_{1}*R*^{k}_{2}*,* *R*_{1}^{k}

*R*^{k}_{2}*R*^{k}_{3}*,*
)

in Theorem 1 and 2

Corollary 2.6. Let be M an interior point of triangle*ABC.* Then for every real number*k∈*[0,1]we
have

a).

*R*^{k}_{a}*R*^{2k}_{2} + *R*^{k}_{b}

*R*^{2k}_{3} + *R*^{k}_{c}*R*^{2k}_{1} *≥* 1

*R*_{1}* ^{k}* + 1

*R*

^{k}_{2}+ 1

*R*^{k}_{3}
b).

1

*R*^{k}_{3}

√
*R*^{k}_{b}*R*^{k}_{c}

+ 1

*R*^{k}_{1}√

*R*^{k}_{c}*R*^{k}* _{a}* + 1

*R*

_{2}

^{k}√
*R*^{k}_{a}*R*^{k}_{b}

*≤* 1
*R*^{2k}_{1} + 1

*R*^{2k}_{2} + 1
*R*^{2k}_{3}

*Proof.* We take in Theorem 1:

(λ1*, λ*2*, λ*3) =
( 1

*R*^{2k}_{2} *,* 1
*R*^{2k}_{3} *,* 1

*R*^{2k}_{1}
)

Theorem 3. Let be*M* a interior point of triangle*ABC.* Then for every real number*k∈*[0,1]we have
a).

*R**a**R**b**R**c*

*R*1*R*2*R*3 *≥* 1
8^{1}^{k}*·*

∏ (*b** ^{k}*+

*c*

*)1/k*

^{k}*abc* *≥*1
b).

∏ (*R*^{k}_{3}
*R*^{k}* _{b}* +

*R*

^{k}_{2}

*R*^{k}* _{c}*
)

*≤* 64 (abc)^{k}

∏(b* ^{k}*+

*c*

*)*

^{k}*≤*8

*Proof.* From the inequalities*R*1*≥* ^{c}_{a}*r*2+_{a}^{b}*r*3*, R*1*≥* _{a}^{b}*r*2+^{c}_{a}*r*3 and(x+*y)*^{k}*≥*2^{k}^{−}^{1}(

*x** ^{k}*+

*y*

*)*

^{k}*,*if

*k∈*[0,1]it result.

*R*^{k}_{1}*≥*
(*b*

*ar*2+ *c*
*ar*3

)*k*

*≥*2^{k}^{−}^{1}
(*b*^{k}

*a*^{k}*r*^{k}_{2}+*c*^{k}*a*^{k}*r*_{3}^{k}

)

and

*R*^{k}_{1}*≥*
(*c*

*ar*2+ *b*
*ar*3

)*k*

*≥*2^{k}^{−}^{1}
(*c*^{k}

*a*^{k}*r*^{k}_{2}+*b*^{k}*a*^{k}*r*_{3}^{k}

)

Adding the last two inequalities we obtain

2R^{k}_{1} *≥*2^{k}^{−}^{1}
[

*r*^{k}_{2}

(*b** ^{k}*+

*c*

^{k}*a*

^{k})
+*r*^{k}_{3}

(*b** ^{k}*+

*c*

^{k}*a*

^{k})]

= 2^{k}^{−}^{1}(

*r*_{2}* ^{k}*+

*r*

^{k}_{3}) (

*b** ^{k}*+

*c*

*)*

^{k}*a*

^{k}and the similar inequalities.

By multiplication we obtain:

8 (R1*R*2*R*3)^{k}*≥*8^{k}^{−}^{1}

∏ (*r*_{2}* ^{k}*+

*r*

^{k}_{3}) (

*b** ^{k}*+

*c*

*)*

^{k}*a*

^{k}*b*

^{k}*c*

*Taking account by equalities*

^{k}*r*2=

^{R}_{2R}

^{3}

^{R}^{1}

*b* *, r*3= ^{R}_{2R}^{1}^{R}^{2}

*c* we obtain:

8 (R1*R*2*R*3)^{k}*≥*8^{k}^{−}^{1}∏ (*R*^{k}_{3}*R*_{1}^{k}

2^{k}*R*_{b}* ^{k}* +

*R*

^{k}_{1}

*R*

^{k}_{2}2

^{k}*R*

^{k}

_{c}) ∏ (

*b** ^{k}*+

*c*

*)*

^{k}*a*

^{k}*b*

^{k}*c*

*or*

^{k}64*≥*∏ (*R*^{k}_{3}
*R*^{k}* _{b}* +

*R*

_{2}

^{k}*R*_{c}* ^{k}*
) ∏ (

*b** ^{k}*+

*c*

*)*

^{k}*a*^{k}*b*^{k}*c*^{k}*≥*8∏ (*R*^{k}_{3}
*R*^{k}* _{b}* +

*R*

^{k}_{2}

*R*^{k}* _{c}*
)

or

∏ (R^{k}_{3}
*R*^{k}* _{b}* +

*R*

^{k}_{2}

*R*^{k}* _{c}*
)

*≤* 64a^{k}*b*^{k}*c*^{k}

∏(b* ^{k}*+

*c*

*)*

^{k}*≤*8

b). We have

64*≥*∏ (*R*^{k}_{3}
*R*^{k}* _{b}* +

*R*

^{k}_{2}

*R*^{k}* _{c}*
) ∏ (

*b** ^{k}*+

*c*

*)*

^{k}*a*

^{k}*b*

^{k}*c*

^{k}*≥*8

∏ (*b** ^{k}*+

*c*

*)*

^{k}*a*^{k}*b*^{k}*c*^{k}*·* (R1*R*2*R*3)* ^{k}*
(R

*a*

*R*

*b*

*R*

*c*)

^{k}*.*