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 1 of 5: Introduce the second tangent from NN
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NX tangent to ω at X (X≠E),M′=NX∩ABNX \text{ tangent to } \omega \text{ at } X\ (X\ne E),\quad M'=NX\cap AB
Detailed analysis

Let the tangent line from NN to ω\omega other than NENE touch ω\omega at X≠EX\ne E and meet line ABAB at M′M'. It suffices to show M′R∥ACM'R\parallel AC, since this forces M′=MM'=M and would make MNMN the constructed tangent line.