HYACINTHOS 7185

Dear friends,

I have the following three results:
1.If J1,J2,J3 are incenters of triangles
IaBC,IbCA,IcAB(where Ia,Ib,Ic are
excenters of triangle ABC),and
BJ3/\CJ2=K1,CJ1/\AJ3=K2,AJ2/\BJ1=K3
then AK1,BK2,CK3 are concurent in
point Q=X(1130) given in barycentrics by
Q(a(2sinA/2+1):b(2sinB/2+1):c(2sinC/2+1)).
2.If I is incenter of ABC and
IJ1/\BC=A1,IJ2/\CA=B1,IJ3/\AB=C1 then
AA1,BB1,CC1 are concurent in point S,
given in barycentrics by
S(a/(1+sinA/2):b/(1+sinB/2):c/(1+sinC/2)).
3.If I1,I2,I3 are incenters of triangles IBC,ICA,IAB
and J1I1/\BC=V1,J2I2/\CA=V2,J3I3/\AB=V3
then AV1,BV2,CV3 are concurent in point
V, given in barycentrics by
V(cosA/2+sinA/2-1:cosB/2+sinB/2-1:
:cosC/2+sinC/2-1).

Best regards
Sincerely
Milorad R.Stevanovic