MathDB

{(a, b, c) | ab + bc + ca> 0} when a <k (b + c), b <k (c + a) , c <k (a + b)

Source: Mathley 2014.1 p4

8/19/2020

Let

S_k

be the set of all triplets of real numbers

(a, b, c)

satisfying

a <k (b + c)

,

b <k (c + a)

, and

c <k (a + b)

. For what value of

k

then

S_k

is a subset of

\{(a, b, c) | ab + bc + ca> 0\}

?
Michel Bataille, France

algebrainequalities

common tangent of 3 circles wanted, mixtilinear circles related

Source: Mathley 2014.2 p4

8/20/2020

Let

(O)

be the circumcircle of triangle

ABC

, and

P

a point on the arc

BC

not containing

A

.

(Q)

is the

A

-mixtilinear circle of triangle

ABC

, and

(K), (L)

are the

P

-mixtilinear circles of triangle

PAB, PAC

respectively. Prove that there is a line tangent to all the three circles

(Q), (K)

and

(L)

.
Nguyen Van Linh, a student at Hanoi Foreign Trade University Cabinet

geometrycommon tangentmixtilinear

perpendicular wanted, reflections of B,C across AC,AB , circumcircle

Source: Mathley 2014.3 p4

8/18/2020

Let

ABC

be an acute triangle with

E, F

being the reflections of

B,C

about the line

AC, AB

respectively. Point

D

is the intersection of

BF

and

CE

. If

K

is the circumcircle of triangle

DEF

, prove that

AK

is perpendicular to

BC

.
Nguyen Minh Ha, College of Pedagogical University of Hanoi

geometrygeometric transformationreflectioncircumcircle

MN passes through circumcenter of ABC, BE = AE, AF = CF, orthocenter

Source: Mathley 2015 p4

8/18/2020

Points

E, F

are in the plane of triangle

ABC

so that triangles

ABE

and

ACF

are the opposite directed, and the two triangles are isosceles in that

BE = AE, AF = CF

. Let

H, K

be the orthocenter of triangle

ABE, ACF

respectively. Points

M, N

are the intersections of

BE

and

CF, CK

and

CH

. Prove that

MN

passes through the center of the circumcircle of triangle

ABC

.
Nguyen Minh Ha, High School for Education, Hanoi Pedagogical University

Circumcenterequal segmentsorthocenterisoscelestrigonometry

4

Problems(4)