Problem 1

Let

ABC

be a triangle with

\angle A<60^\circ

. Let

X

and

Y

be the points on the sides

AB

and

AC

, respectively, such that

CA+AX=CB+BX

and

BA+AY=BC+CY

. Let

P

be the point in the plane such that the lines

PX

and

PY

are perpendicular to

AB

and

AC

, respectively. Prove that

\angle BPC<120^\circ

Step 2 of 5: Use the midpoint of IP

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M=\operatorname{mid}(DX)=\operatorname{mid}(AB),\quad N=\operatorname{mid}(EY)=\operatorname{mid}(AC)

Detailed analysis

Let O be the midpoint of IP, and let M,N be the perpendicular feet from O to AB,AC. Because P lies on the perpendicular through X to AB and I lies on the perpendicular through D, M is the midpoint of DX. Similarly N is the midpoint of EY. The equalities BD=AX and CE=AY therefore make M and N the midpoints of AB and AC.