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Let

I

be the incenter of the non-isosceles triangle

ABC

and let

A',B',C'

be the tangency points of the incircle with the sides

BC,CA,AB

respectively. The lines

AA'

and

BB'

intersect in

P

, the lines

AC

and

A'C'

M

and the lines

B'C'

and

BC

intersect in

N

. Prove that the lines

IP

and

MN

are perpendicular. Alternative formulation. The incircle of a non-isosceles triangle

ABC

has center

I

and touches the sides

BC

CA

and

AB

A^{\prime}

B^{\prime}

and

C^{\prime}

, respectively. The lines

AA^{\prime}

and

BB^{\prime}

intersect in

P

, the lines

AC

and

A^{\prime}C^{\prime}

intersect in

M

, and the lines

BC

and

B^{\prime}C^{\prime}

intersect in

N

. Prove that the lines

IP

and

MN

are perpendicular.

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