MathDB

Let

ABC

be an acute nonisosceles triangle with incenter

I

and

(d)

is an arbitrary line tangent to

(I)

K

. The lines passes through

I

, perpendicular to

IA, IB, IC

cut

(d)

A_1, B_1,C_1

respectively. Suppose that

(d)

cuts

BC, CA, AB

M,N, P

respectively. The lines through

M,N,P

and respectively parallel to the internal bisectors of

A, B, C

in triangle

ABC

meet each other to define a triange

XYZ

. Prove that three lines

AA_1, BB_1, CC_1

are concurrent and

IK

is tangent to the circle

(XY Z)

Problem Statement