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Classic geo

Source: 2007 Bulgarian Autumn Math Competition, Problem 10.2

March 17, 2022

geometryCyclicangle bisector

Problem Statement

Let

AC>BC

in

\triangle ABC

and

M

and

N

be the midpoints of

I

and

BC

respectively. The angle bisector of

\angle B

intersects

\overline{MN}

at

P

. The incircle of

\triangle ABC

has center

S=AB\cap RN

and touches

BC

at

Q

. The perpendiculars from

P

and

Q

to

MN

and

BC

respectively intersect at

R

. Let

S=AB\cap RN

. a) Prove that

PCQI

is cyclic b) Express the length of the segment

BS

with

a

,

b

,

c

- the side lengths of

\triangle ABC

.

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