MathLabs

Problem 3

Let ABAB and ACAC be two distinct rays not lying on the same line, and let ω\omega be a circle with center OO that is tangent to ray ACAC at EE and ray ABAB at FF. Let RR be a point on segment EFEF. The line through OO parallel to EFEF intersects line ABAB at PP. Let NN be the intersection of lines PRPR and ACAC, and let MM be the intersection of line ABAB and the line through RR parallel to ACAC. Prove that line MNMN is tangent to ω\omega.
Step 2 of 5: Auxiliary circle and a first similarity
Previous stepNext step
Q=PO∩AC, Y=PO∩⊙(AM′O),∠AYM′=∠AOM′=90∘−∠M′OP,∠EOQ=∠FOP=∠M′AO=∠M′YPQ=PO\cap AC,\ Y=PO\cap\odot(AM'O),\quad \angle AYM'=\angle AOM'=90^\circ-\angle M'OP,\quad \angle EOQ=\angle FOP=\angle M'AO=\angle M'YP
Detailed analysis

Let Q=PO∩ACQ=PO\cap AC and let YY be the second intersection of line POPO with the circumcircle of △AM′O\triangle AM'O; note ∠AOM′=90∘−∠M′OP\angle AOM'=90^\circ-\angle M'OP since OAOA bisects the angle between the two tangent radii. Angle chasing around OO gives ∠EOQ=∠FOP=∠M′AO=∠M′YP\angle EOQ=\angle FOP=\angle M'AO=\angle M'YP, and symmetrically ∠EQO=∠M′PY\angle EQO=\angle M'PY, so △M′YP∼△EOQ\triangle M'YP\sim\triangle EOQ.