Let $ABCD$ be a parallelogram and $M$ is the intersection of $AC$ and $BD$. The point $N$ is inside of the $\triangle AMB$ such that $\angle AND=\angle BNC$. Prove that $\angle MNC=\angle NDA$ and $\angle MND=\angle NCB$.
Problem
Source: Rioplatense L-2 2022 #4
Tags: geometry, parallelogram
teoira
13.06.2023 19:22
1. $\sigma_M: N\to N'$, $\angle AND\to \angle CN'B \Longrightarrow \angle CN'B=\angle AND=\alpha$; $\angle BNC\to \angle AN'D \Longrightarrow \angle BNC=\angle AN'D=\alpha$ 2. $\sigma_M: \Delta ADN\to \Delta BN'C\Longrightarrow \angle ADN=\angle CBN'$ 3. $\sigma_M: \Delta BNC\to \Delta AN'D\Longrightarrow \angle BCN=\angle DAN'$ 4. $ANN'D$ -- cyclic $\Longrightarrow \angle DNO=\angle DAN'=\angle BCN$ 5. $BCN'N$ -- cyclic $\Longrightarrow \angle CNO=\angle CBN'=\angle ADN$ [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(12.989429175475685cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = 2.49291562560062, xmax = 18.729702094945225, ymin = -7.992961752834888, ymax = 5.064120699596381; /* image dimensions */ pen ffttww = rgb(1.,0.2,0.4); pen qqzzqq = rgb(0.,0.6,0.); draw(arc((9.145093414960112,-5.746981317007977),0.507399577167019,-24.17856938658694,59.22039532338483)--(9.145093414960112,-5.746981317007977)--cycle, linewidth(2.) + ffttww); draw(arc((9.145093414960112,-5.746981317007977),0.507399577167019,107.79652156154411,191.5568795945546)--(9.145093414960112,-5.746981317007977)--cycle, linewidth(2.) + ffttww); draw(arc((8.818742921585443,2.7405868159185527),0.507399577167019,-72.20076845149119,11.564968369905614)--(8.818742921585443,2.7405868159185527)--cycle, linewidth(2.) + ffttww); draw(arc((8.818742921585443,2.7405868159185527),0.507399577167019,155.85152874331786,239.22285728118027)--(8.818742921585443,2.7405868159185527)--cycle, linewidth(2.) + ffttww); draw(arc((9.145093414960112,-5.746981317007977),1.1839323467230443,92.20196161589588,107.79652156154411)--(9.145093414960112,-5.746981317007977)--cycle, linewidth(2.) + blue); draw(arc((14.947640122440168,3.9947640122440173),1.1839323467230443,-120.7796046766152,-105.2551187030578)--(14.947640122440168,3.9947640122440173)--cycle, linewidth(2.) + blue); draw(arc((11.947640122440168,-7.005235987755983),1.014799154334038,74.74488129694222,107.79923154850883)--(11.947640122440168,-7.005235987755983)--cycle, linewidth(2.) + qqzzqq); draw(arc((6.017453292519943,3.9965093414960116),1.014799154334038,-105.25973545128168,-72.20347843845592)--(6.017453292519943,3.9965093414960116)--cycle, linewidth(2.) + qqzzqq); draw(arc((3.017453292519943,-7.),1.1839323467230443,59.22285728118025,74.74026454871832)--(3.017453292519943,-7.)--cycle, linewidth(2.) + blue); draw(arc((9.145093414960112,-5.746981317007977),1.014799154334038,59.22039532338483,92.20196161589587)--(9.145093414960112,-5.746981317007977)--cycle, linewidth(2.) + qqzzqq); /* draw figures */ draw((3.017453292519943,-7.)--(11.947640122440168,-7.005235987755983), linewidth(2.)); draw((11.947640122440168,-7.005235987755983)--(14.947640122440168,3.9947640122440173), linewidth(2.)); draw((14.947640122440168,3.9947640122440173)--(6.017453292519943,3.9965093414960116), linewidth(2.)); draw((6.017453292519943,3.9965093414960116)--(3.017453292519943,-7.), linewidth(2.)); draw((3.017453292519943,-7.)--(14.947640122440168,3.9947640122440173), linewidth(2.)); draw((6.017453292519943,3.9965093414960116)--(11.947640122440168,-7.005235987755983), linewidth(2.)); draw((6.017453292519943,3.9965093414960116)--(9.145093414960112,-5.746981317007977), linewidth(2.)); draw((9.145093414960112,-5.746981317007977)--(3.017453292519943,-7.), linewidth(2.)); draw((9.145093414960112,-5.746981317007977)--(11.947640122440168,-7.005235987755983), linewidth(2.)); draw((9.145093414960112,-5.746981317007977)--(14.947640122440168,3.9947640122440173), linewidth(2.)); draw((6.017453292519943,3.9965093414960116)--(8.818742921585443,2.7405868159185527), linewidth(2.)); draw((8.818742921585443,2.7405868159185527)--(3.017453292519943,-7.), linewidth(2.)); draw((8.818742921585443,2.7405868159185527)--(11.947640122440168,-7.005235987755983), linewidth(2.)); draw((8.818742921585443,2.7405868159185527)--(14.947640122440168,3.9947640122440173), linewidth(2.)); draw(circle((12.80006227168441,-1.3563880358899219), 5.712802001104499), linewidth(2.)); draw((8.818742921585443,2.7405868159185527)--(9.145093414960112,-5.746981317007977), linewidth(2.)); draw((22.63667883913127,-2.6314395541033955)--(26.019342686911397,-2.6314395541033955), linewidth(4.)); draw((3.017453292519943,-7.)--(9.145093414960112,-5.746981317007977), linewidth(2.)); draw((9.145093414960112,-5.746981317007977)--(6.017453292519943,3.9965093414960116), linewidth(2.)); draw((6.017453292519943,3.9965093414960116)--(3.017453292519943,-7.), linewidth(2.)); draw((14.947640122440168,3.9947640122440173)--(11.947640122440168,-7.005235987755983), linewidth(2.)); draw((11.947640122440168,-7.005235987755983)--(9.145093414960112,-5.746981317007977), linewidth(2.)); draw((9.145093414960112,-5.746981317007977)--(14.947640122440168,3.9947640122440173), linewidth(2.)); draw((14.947640122440168,3.9947640122440173)--(11.947640122440168,-7.005235987755983), linewidth(2.)); draw((6.017453292519943,3.9965093414960116)--(3.017453292519943,-7.), linewidth(2.)); /* dots and labels */ dot((3.017453292519943,-7.),dotstyle); label("$A$", (2.7635287334230303,-7.451735537190069), NE * labelscalefactor); dot((11.947640122440168,-7.005235987755983),dotstyle); label("$B$", (11.845981164712668,-7.688522006534677), NE * labelscalefactor); dot((14.947640122440168,3.9947640122440173),dotstyle); label("$C$", (15.008771862387087,4.167714779934649), NE * labelscalefactor); dot((6.017453292519943,3.9965093414960116),dotstyle); label("$D$", (6.078539304247554,4.167714779934649), NE * labelscalefactor); dot((8.981918168272777,-1.5031972505447122),linewidth(4.pt) + dotstyle); label("$O$", (9.292069959638674,-1.6842936767249614), NE * labelscalefactor); dot((9.145093414960112,-5.746981317007977),dotstyle); label("$N$", (9.004543532577364,-6.284716509705927), NE * labelscalefactor); dot((8.818742921585443,2.7405868159185527),dotstyle); label("$N'$", (8.750843743993853,3.2205689025562148), NE * labelscalefactor); dot((24.51593653234245,-2.6314395541033955),dotstyle); label("$n = 6$", (24.328010763021336,-2.377739765519886), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]