Let $ABC$ be an acute triangle with $AB<AC$. Let $D,E,F$ be the feet of the altitudes relatives to the vertices $A,B,C$, respectively. The circumcircle $\Gamma$ of $AEF$ cuts the circumcircle of $ABC$ at $A$ and $M$. Assume that $BM$ is tangent to $\Gamma$. Prove that $M$, $F$ and $D$ are collinear.
Problem
Source:
Tags: geometry
407420
06.12.2022 19:30
Let $H$ be the orthocenter of $ABC$ and let $G=AM\cap BC$. Then $E,F,G$ are collinear by radical axes on $(AEF),(BCEF),(ABC)$. Consider the inversions $(A,\sqrt{AF\cdot AB})$ and $(B,BG)$. Done.
gnoka
31.01.2023 13:33
Let $H$ be the orthocenter of $ABC$ since $A,F,E,H,M$ concyclic, $BM$ is tangent to $\Gamma$, we have $BM^2=BH\cdot BE=BD \cdot BC$ so $\angle BMD=\angle BCM=\angle BAM=\angle BMF$ $\therefore M,F,D $collinear
teoira
11.06.2023 21:53
All that is needed is equality of angles. [asy][asy] import graph; size(7.725271658071063cm); 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 = 1.0037200383402467, xmax = 8.72899169641131, ymin = -7.826859088365633, ymax = 0.341479620683004; /* image dimensions */ pen qqzzqq = rgb(0.,0.6,0.); draw(arc((7.388647928371843,-1.060442736985375),0.3408208084443116,-146.21712923230524,-81.17903161793687)--(7.388647928371843,-1.060442736985375)--cycle, linewidth(2.) + red); draw(arc((6.689274784938306,-6.502173877608809),0.3408208084443116,163.8555044619845,228.89360207635292)--(6.689274784938306,-6.502173877608809)--cycle, linewidth(2.) + red); draw(arc((7.388647928371842,-5.),0.5680346807405193,180.,245.03453607992137)--(7.388647928371842,-5.)--cycle, linewidth(2.) + red); draw(arc((7.388647928371842,-5.),0.3408208084443116,0.,65.03809761436838)--(7.388647928371842,-5.)--cycle, linewidth(2.) + red); /* draw figures */ draw((1.5,-5.)--(8.,-5.), linewidth(1.2)); draw((8.,-5.)--(7.388647928371843,-1.060442736985375), linewidth(1.2)); draw((7.388647928371843,-1.060442736985375)--(1.5,-5.), linewidth(1.2)); draw((1.5,-5.)--(7.847149566870339,-4.015030680460111), linewidth(1.2)); draw((7.388647928371843,-1.060442736985375)--(7.388647928371842,-5.), linewidth(1.2)); draw((5.990275125506986,-1.9959664427559018)--(8.,-5.), linewidth(1.2)); draw(circle((4.444323964185922,-4.543091155966069), 2.979565286721525), linewidth(1.2)); draw(circle((4.75,-3.4871302125266195), 3.584867500180411), linewidth(1.2)); draw((6.689274784938306,-6.502173877608809)--(8.,-5.), linewidth(1.2)); draw((xmin, 1.1460631567525792*xmin-14.168505254020634)--(xmax, 1.1460631567525792*xmax-14.168505254020634), linewidth(2.)); /* line */ draw((xmin, 2.147886134480319*xmin-20.869974437906613)--(xmax, 2.147886134480319*xmax-20.869974437906613), linewidth(2.) + linetype("4 4") + qqzzqq); /* line */ draw((1.5,-5.)--(6.689274784938306,-6.502173877608809), linewidth(1.2)); draw((5.990275125506986,-1.9959664427559018)--(7.388647928371842,-5.), linewidth(1.2)); draw((10.921605564069713,-5.940983948307116)--(13.193744287031791,-5.940983948307116), linewidth(4.)); draw((6.689274784938306,-6.502173877608809)--(7.388647928371842,-5.), linewidth(1.2)); draw((7.388647928371842,-5.)--(7.847149566870339,-4.015030680460111), linewidth(1.2)); /* dots and labels */ dot((1.5,-5.),linewidth(3.pt) + dotstyle); label("$A$", (1.1,-5.191178169729633), NE * labelscalefactor); dot((8.,-5.),linewidth(3.pt) + dotstyle); label("$B$", (8.092792853981928,-5.2138995569592534), NE * labelscalefactor); dot((7.388647928371843,-1.060442736985375),linewidth(3.pt) + dotstyle); label("$C$", (7.433872624322925,-0.9877215322498061), NE * labelscalefactor); dot((7.847149566870339,-4.015030680460111),linewidth(3.pt) + dotstyle); label("$D$", (7.9,-4.1573550507818915), NE * labelscalefactor); dot((5.990275125506986,-1.9959664427559018),linewidth(3.pt) + dotstyle); label("$E$", (5.934261067167954,-1.8284128597457714), NE * labelscalefactor); dot((7.388647928371842,-5.),linewidth(3.pt) + dotstyle); label("$F$", (7.4,-5.36), NE * labelscalefactor); dot((6.689274784938306,-6.502173877608809),linewidth(3.pt) + dotstyle); label("$M$", (6.809034475508354,-6.7362325013438396), NE * labelscalefactor); dot((7.388647928371842,-4.086182311932138),linewidth(3.pt) + dotstyle); label("$H$", (7.433872624322925,-4.021026727404167), NE * labelscalefactor); dot((4.444323964185921,-4.543091155966069),linewidth(3.pt) + dotstyle); dot((12.739316542439376,-5.940983948307116),dotstyle); label("$n = 5$", (12.614348912676462,-5.770573544084961), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]