MathDB

a) The incircle of a triangle

ABC

touches

AC

and

AB

at points

B_0

and

C_0

respectively. The bisectors of angles

B

and

C

meet the perpendicular bisector to the bisector

AL

in points

Q

and

P

respectively. Prove that the lines

PC_0, QB_0

and

BC

concur.
b) Let

AL

be the bisector of a triangle

ABC

. Points

O_1

and

O_2

are the circumcenters of triangles

ABL

and

ACL

respectively. Points

B_1

and

C_1

are the projections of

C

and

B

to the bisectors of angles

B

and

C

respectively. Prove that the lines

O_1C_1, O_2B_1,

and

BC

concur.
c) Prove that the two points obtained in pp. a) and b) coincide.

Problem Statement