MathDB

Let

M

be the point of intersection of the diagonals of a cyclic quadrilateral

ABCD

. Let

I_1

and

I_2

are the incenters of triangles

AMD

and

BMC

, respectively, and let

L

be the point of intersection of the lines

DI_1

and

CI_2

. The foot of the perpendicular from the midpoint

T

I_1I_2

CL

N

, and

F

is the midpoint of

TN

. Let

G

and

J

be the points of intersection of the line

LF

with

I_1N

and

I_1I_2

, respectively. Let

O_1

be the circumcenter of triangle

LI_1J

, and let

\Gamma_1

and

\Gamma_2

be the circles with diameters

O_1L

and

O_1J

, respectively. Let

V

and

S

be the second points of intersection of

I_1O_1

with

\Gamma_1

and

\Gamma_2

, respectively. If

K

is point where the circles

\Gamma_1

and

\Gamma_2

meet again, prove that

K

is the circumcenter of the triangle

SVG

G4