Restore a triangle $ABC$ by the Nagel point, the vertex $B$ and the foot of the altitude from this vertex.
Problem
Source: Sharygin 2018
Tags: geometry
04.04.2018 22:31
LEMMA(WELL KNOWN) In a $\Delta ABC$ let $N$ be the nagel point and $BN\cap AC=L$ and $BN$ meets incircle at $X$ closer to $B$ then $BX=NL$ _______________________________________________ Now let $E$ be the foot of $B-$ altitude now we can easily construct the line $AC$ let $BN\cap AC=L$ then construct a point $X$ on $BN$ such that $NL=BX$ now by lemma $X$ lies on incircle and draw perpendicular through $X$ to $AC$ let it meet $AC$ at $Y$ now circle with diameter $XY$ is the incircle now draw tangents from $B$ to incircle they meet the line $AC$ at $A,C$ so we are done $\blacksquare$
04.04.2018 22:32
This follows from the fact that the isotomic point of the Nagel point on the $B-\text {Nagel} $ cevian is the antipode of the incircle touchpoint with $AC $ in the incircle. For proving this, first note that the Nagel point of a triangle is the anticomplement of the triangle's incenter with respect to the triangle and then a suitable homothety around the centroid finishes the problem.
04.04.2018 22:37
I'm assuming both ruler and compass are allowed. Let $BX$ be the altitude and $N$ the Nagel point. Construct line $\ell$ passing through $X$ perpendicular to $BX$ and intersect it with line $BN$ to obtain point $L$. Reflect $N$ in the mid-point of $BL$. By USAMO 2001/2, it is clear that the image $Y$ is the diameter of the incircle. Draw line through $Y$ perpendicular to $\ell$ and intersect it with the line joining the mid-point of $BX$ to $L$; thus obtaining the incenter $I$ of $\triangle ABC$. Drop feet $D$ from $I$ on $\ell$. The midpoint $M$ of $DL$ is the midpoint of the line $AC$ from the beta timeline, hence we obtain the perpendicular bisector of $AC$. Now we construct the internal/external bisectors of angle $ABC$ and intersect them with the perpendicular bisector of $AC$ to get the mid-point of arcs. Intersect circumcircle with $\ell$ to get $A,C$.
04.04.2018 22:38
USAMO 2001/02 kills it. (Note that in the USAMO Q, $P$ is the Nagel Point)
04.04.2018 22:40
Nagel point of the medial triangle is the incenter of the triangle. Construct the triangle using this and then construct $ABC$ which is the medial triangle.
05.04.2018 06:13
It is already there in Egmo.(problem 7.39) just to notice $P$ is nagel point
05.04.2018 07:38
Let $D$ be foot of altitude and $N$ be Nagel point. Construct line $\ell$ perpendicular to $BD$. Clearly, $A,C$ lie on $\ell$. Let $BN$ meet $\ell$ at $E$. Construct centroid $G$ of triangle $BDE$ and draw line through $G$ parallel to $\ell$, let this line meet $BN$ at $X$. Let $M$ be midpoint of $BD$ and $N'$ be point on $BN$ such that $N,X,N'$ lie in that order and $NX=2N'X$. Its well known that incenter of $\triangle ABC$ lies on $ME$. Now, centroid of $\triangle ABC$ lies on $BX$ and its well known that $N$, incenter and centroid are collinear and line joining them is divided in $2:1$ ratio by centroid. Thus, by similarity incenter lies on line parallel to $N'$ parallel to $\ell$. So, construct a line through $N'$ parallel to $\ell$ and let it meet $ME$ at $I$. $I$ becomes incenter. Let $J$ be the foot of $I$ on $\ell$. Construct circle with center $I$ and radius $IJ$. Tangents to this circle from $B$ meet $\ell$ at $A$ and $C$.
05.04.2018 08:19
05.04.2018 09:13
We can draw line $\ell$, which is line $AC$. Let $BN_a$ meet this line at $E$, and let $D$ be the foot of the altitude. Using Van Aubel's relation, we get $\frac{BN_a}{N_aE} = \frac{b}{s-b}$. But note that the $B-$ exradius is $\frac{\Delta}{s-b}$. Hence $\frac{BN_a}{N_aE} = \frac{r_b}{2h_a}$. We know $h_a,BN_a,N_aE$. Hence we can find $r_b$ (try to use power of point). So, we can find the $B-$excenter as we know $E$. Now we can draw the excircle and then we can draw tangents from $B$ to this circle, which meet the line $\ell$ at $A,C$. $\blacksquare$.
05.04.2018 09:43
24.04.2018 08:28
Key Lemma: USAMO 2001 Problem 2 [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5cm); 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 = -2.9616084159818032, xmax = 14.965664311290897, ymin = -3.5619027101090848, ymax = 6.983551835345453; /* image dimensions */ /* draw figures */ /* dots and labels */ dot((2.0667220000000026,4.28964),linewidth(4pt) + dotstyle); label("$B$", (2.1474824931090986,4.438097289890909), NE * labelscalefactor); dot((2.1,-1.6),linewidth(4pt) + dotstyle); label("$D$", (2.165664311290917,-1.4528118010181772), NE * labelscalefactor); dot((6.24,-0.06),linewidth(4pt) + dotstyle); label("$N$", (6.311118856745456,0.09264274443636698), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy] [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5cm); 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 = -1.140557941287453, xmax = 12.32846739851172, ymin = -2.7643615585484005, ymax = 5.158594523686413; /* image dimensions */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); draw((2.389760830721481,-1.6031606039532174)--(2.3929214346746983,-1.3133997732317364)--(2.103160603953217,-1.3102391692785194)--(2.1,-1.6)--cycle, linewidth(1.6)); /* draw figures */ draw((2.165227182679484,4.379959184570233)--(2.1,-1.6), linewidth(2) + wrwrwr); draw((2.1,-1.6)--(7.709488760476195,-1.6611862283378427), linewidth(2) + wrwrwr); draw((7.709488760476195,-1.6611862283378427)--(2.165227182679484,4.379959184570233), linewidth(2) + wrwrwr); /* dots and labels */ dot((2.165227182679484,4.379959184570233),linewidth(4pt) + dotstyle); label("$B$", (2.2198682591086896,4.489241337428645), NE * labelscalefactor); dot((2.1,-1.6),linewidth(4pt) + dotstyle); label("$D$", (2.1515669135721827,-1.49395653156937), NE * labelscalefactor); dot((6.24,-0.06),linewidth(4pt) + dotstyle); label("$N$", (6.290628453084505,0.04965387755568858), NE * labelscalefactor); dot((7.709488760476195,-1.6611862283378427),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy] [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5cm); 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 = -1.140557941287453, xmax = 12.32846739851172, ymin = -2.7643615585484005, ymax = 5.158594523686413; /* image dimensions */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen qqwwtt = rgb(0,0.4,0.2); draw((2.389760830721481,-1.6031606039532174)--(2.3929214346746983,-1.3133997732317364)--(2.103160603953217,-1.3102391692785194)--(2.1,-1.6)--cycle, linewidth(1.6)); /* draw figures */ draw((2.165227182679484,4.379959184570233)--(2.1,-1.6), linewidth(2) + wrwrwr); draw((2.1,-1.6)--(7.709488760476195,-1.6611862283378427), linewidth(2) + wrwrwr); draw((2.165227182679484,4.379959184570233)--(3.63471594315568,2.7787729562323893), linewidth(2) + qqwwtt); draw((2.960357308647863,3.6347850922946834)--(2.8395858171873014,3.5239470485079396), linewidth(2) + qqwwtt); draw((3.63471594315568,2.7787729562323893)--(6.24,-0.06), linewidth(2) + wrwrwr); draw((6.24,-0.06)--(7.709488760476195,-1.6611862283378427), linewidth(2) + qqwwtt); draw((7.035130125968377,-0.8051740922755496)--(6.914358634507817,-0.9160121360622935), linewidth(2) + qqwwtt); /* dots and labels */ dot((2.165227182679484,4.379959184570233),linewidth(4pt) + dotstyle); dot((2.1,-1.6),linewidth(4pt) + dotstyle); dot((6.24,-0.06),linewidth(4pt) + dotstyle); dot((7.709488760476195,-1.6611862283378427),linewidth(4pt) + dotstyle); dot((3.63471594315568,2.7787729562323893),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy] [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5cm); 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 = -1.140557941287453, xmax = 12.32846739851172, ymin = -2.7643615585484005, ymax = 5.158594523686413; /* image dimensions */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen qqwwtt = rgb(0,0.4,0.2); draw((2.389760830721481,-1.6031606039532174)--(2.3929214346746983,-1.3133997732317364)--(2.103160603953217,-1.3102391692785194)--(2.1,-1.6)--cycle, linewidth(1.6)); draw((3.8765378468977474,-1.6193778176572868)--(3.879698450850965,-1.3296169869358059)--(3.5899376201294837,-1.3264563829825888)--(3.5867770161762667,-1.6162172137040696)--cycle, linewidth(1.6)); /* draw figures */ draw((2.165227182679484,4.379959184570233)--(2.1,-1.6), linewidth(2) + wrwrwr); draw((2.1,-1.6)--(7.709488760476195,-1.6611862283378427), linewidth(2) + wrwrwr); draw((2.165227182679484,4.379959184570233)--(3.63471594315568,2.7787729562323893), linewidth(2) + qqwwtt); draw((2.960357308647863,3.6347850922946834)--(2.8395858171873014,3.5239470485079396), linewidth(2) + qqwwtt); draw((3.63471594315568,2.7787729562323893)--(6.24,-0.06), linewidth(2) + wrwrwr); draw((6.24,-0.06)--(7.709488760476195,-1.6611862283378427), linewidth(2) + qqwwtt); draw((7.035130125968377,-0.8051740922755496)--(6.914358634507817,-0.9160121360622935), linewidth(2) + qqwwtt); draw((3.63471594315568,2.7787729562323893)--(3.6107464796659734,0.5812778712641599), linewidth(2) + wrwrwr); draw((3.7050604314889832,1.7132801013406567)--(3.541146952828658,1.7150680089310284), linewidth(2) + wrwrwr); draw((3.7043154699929945,1.644982818565521)--(3.54040199133267,1.646770726155893), linewidth(2) + wrwrwr); draw((3.6107464796659734,0.5812778712641599)--(3.5867770161762667,-1.6162172137040696), linewidth(2) + wrwrwr); draw((3.6810909679992774,-0.484214983627573)--(3.5171774893389522,-0.4824270760372012), linewidth(2) + wrwrwr); draw((3.6803460065032887,-0.5525122664027083)--(3.5164325278429636,-0.5507243588123366), linewidth(2) + wrwrwr); /* dots and labels */ dot((2.165227182679484,4.379959184570233),linewidth(4pt) + dotstyle); dot((2.1,-1.6),linewidth(4pt) + dotstyle); dot((6.24,-0.06),linewidth(4pt) + dotstyle); dot((7.709488760476195,-1.6611862283378427),linewidth(4pt) + dotstyle); dot((3.63471594315568,2.7787729562323893),linewidth(4pt) + dotstyle); dot((3.5867770161762667,-1.6162172137040696),linewidth(4pt) + dotstyle); dot((3.6107464796659734,0.5812778712641599),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy] [asy][asy]/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(5cm); 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 = -1.140557941287453, xmax = 12.32846739851172, ymin = -2.7643615585484005, ymax = 5.158594523686413; /* image dimensions */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen qqwwtt = rgb(0,0.4,0.2); draw((2.389760830721481,-1.6031606039532174)--(2.3929214346746983,-1.3133997732317364)--(2.103160603953217,-1.3102391692785194)--(2.1,-1.6)--cycle, linewidth(1.6)); draw((3.8765378468977474,-1.6193778176572868)--(3.879698450850965,-1.3296169869358059)--(3.5899376201294837,-1.3264563829825888)--(3.5867770161762667,-1.6162172137040696)--cycle, linewidth(1.6)); /* draw figures */ draw((2.165227182679484,4.379959184570233)--(2.1,-1.6), linewidth(2) + wrwrwr); draw((2.1,-1.6)--(7.709488760476195,-1.6611862283378427), linewidth(2) + wrwrwr); draw((2.165227182679484,4.379959184570233)--(3.63471594315568,2.7787729562323893), linewidth(2) + qqwwtt); draw((2.960357308647863,3.6347850922946834)--(2.8395858171873014,3.5239470485079396), linewidth(2) + qqwwtt); draw((3.63471594315568,2.7787729562323893)--(6.24,-0.06), linewidth(2) + wrwrwr); draw((6.24,-0.06)--(7.709488760476195,-1.6611862283378427), linewidth(2) + qqwwtt); draw((7.035130125968377,-0.8051740922755496)--(6.914358634507817,-0.9160121360622935), linewidth(2) + qqwwtt); draw((3.63471594315568,2.7787729562323893)--(3.6107464796659734,0.5812778712641599), linewidth(2) + wrwrwr); draw((3.7050604314889832,1.7132801013406567)--(3.541146952828658,1.7150680089310284), linewidth(2) + wrwrwr); draw((3.7043154699929945,1.644982818565521)--(3.54040199133267,1.646770726155893), linewidth(2) + wrwrwr); draw((3.6107464796659734,0.5812778712641599)--(3.5867770161762667,-1.6162172137040696), linewidth(2) + wrwrwr); draw((3.6810909679992774,-0.484214983627573)--(3.5171774893389522,-0.4824270760372012), linewidth(2) + wrwrwr); draw((3.6803460065032887,-0.5525122664027083)--(3.5164325278429636,-0.5507243588123366), linewidth(2) + wrwrwr); draw(circle((3.6107464796659734,0.5812778712641599), 2.1976258061006453), linewidth(2) + wrwrwr); /* dots and labels */ dot((2.165227182679484,4.379959184570233),linewidth(4pt) + dotstyle); dot((2.1,-1.6),linewidth(4pt) + dotstyle); dot((6.24,-0.06),linewidth(4pt) + dotstyle); dot((7.709488760476195,-1.6611862283378427),linewidth(4pt) + dotstyle); dot((3.63471594315568,2.7787729562323893),linewidth(4pt) + dotstyle); dot((3.5867770161762667,-1.6162172137040696),linewidth(4pt) + dotstyle); dot((3.6107464796659734,0.5812778712641599),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy] [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki, go to User:Azjps/geogebra */ import graph; size(15cm); 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 = -1.140557941287453, xmax = 12.32846739851172, ymin = -2.7643615585484005, ymax = 5.158594523686413; /* image dimensions */ pen wrwrwr = rgb(0.3803921568627451,0.3803921568627451,0.3803921568627451); pen qqwwtt = rgb(0,0.4,0.2); draw((2.389760830721481,-1.6031606039532174)--(2.3929214346746983,-1.3133997732317364)--(2.103160603953217,-1.3102391692785194)--(2.1,-1.6)--cycle, linewidth(1.6)); draw((3.8765378468977474,-1.6193778176572868)--(3.879698450850965,-1.3296169869358059)--(3.5899376201294837,-1.3264563829825888)--(3.5867770161762667,-1.6162172137040696)--cycle, linewidth(1.6)); /* draw figures */ draw((2.165227182679484,4.379959184570233)--(2.1,-1.6), linewidth(2) + wrwrwr); draw((2.165227182679484,4.379959184570233)--(3.63471594315568,2.7787729562323893), linewidth(2) + qqwwtt); draw((2.960357308647863,3.6347850922946834)--(2.8395858171873014,3.5239470485079396), linewidth(2) + qqwwtt); draw((3.63471594315568,2.7787729562323893)--(6.24,-0.06), linewidth(2) + wrwrwr); draw((6.24,-0.06)--(7.709488760476195,-1.6611862283378427), linewidth(2) + qqwwtt); draw((7.035130125968377,-0.8051740922755496)--(6.914358634507817,-0.9160121360622935), linewidth(2) + qqwwtt); draw((3.63471594315568,2.7787729562323893)--(3.6107464796659734,0.5812778712641599), linewidth(2) + wrwrwr); draw((3.7050604314889832,1.7132801013406567)--(3.541146952828658,1.7150680089310284), linewidth(2) + wrwrwr); draw((3.7043154699929945,1.644982818565521)--(3.54040199133267,1.646770726155893), linewidth(2) + wrwrwr); draw((3.6107464796659734,0.5812778712641599)--(3.5867770161762667,-1.6162172137040696), linewidth(2) + wrwrwr); draw((3.6810909679992774,-0.484214983627573)--(3.5171774893389522,-0.4824270760372012), linewidth(2) + wrwrwr); draw((3.6803460065032887,-0.5525122664027083)--(3.5164325278429636,-0.5507243588123366), linewidth(2) + wrwrwr); draw(circle((3.6107464796659734,0.5812778712641599), 2.1976258061006453), linewidth(2) + wrwrwr); draw((2.165227182679484,4.379959184570232)--(xmax, -0.7382171304353301*xmax + 5.978366982108455), linewidth(2) + wrwrwr); /* ray */ draw((2.165227182679484,4.379959184570233)--(xmin, 4.746246850911421*xmin-5.896743512730076), linewidth(2) + wrwrwr); /* ray */ draw((xmin, -0.010907630080115971*xmin-1.5770939768317567)--(xmax, -0.010907630080115971*xmax-1.5770939768317567), linewidth(2) + wrwrwr); /* line */ /* dots and labels */ dot((2.165227182679484,4.379959184570233),linewidth(4pt) + dotstyle); dot((2.1,-1.6),linewidth(4pt) + dotstyle); dot((6.24,-0.06),linewidth(4pt) + dotstyle); dot((7.709488760476195,-1.6611862283378427),linewidth(4pt) + dotstyle); dot((3.63471594315568,2.7787729562323893),linewidth(4pt) + dotstyle); dot((3.5867770161762667,-1.6162172137040696),linewidth(4pt) + dotstyle); dot((3.6107464796659734,0.5812778712641599),linewidth(4pt) + dotstyle); dot((0.9080322182427787,-1.586998456369176),linewidth(4pt) + dotstyle); dot((10.38823355840968,-1.6904049856727366),linewidth(4pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy][/asy]