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 5 of 5: Conclude M′=MM'=M
Previous stepNext step
M′R∥AC  ⟹  M′=M  ⟹  MN tangent to ωM'R\parallel AC \implies M'=M \implies MN \text{ tangent to } \omega
Detailed analysis

Since M′R∥ACM'R\parallel AC and MM was defined as the intersection of ABAB with the line through RR parallel to ACAC, we must have M′=MM'=M. As line NM′NM' was constructed tangent to ω\omega, line MNMN is tangent to ω\omega, as required.