locus of midpoints wanted, circle related

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Source: 2014 Belarus TST 1.2

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30629 posts

#1 h Dec 29, 2020, 8:16 PM

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Given a triangle $ABC$ . Let $S$ be the circle passing through $C$ , centered at $A$ . Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$ , when $X$ moves along $S$ .

(I. Gorodnin)

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126 posts

#2 h Dec 29, 2020, 8:28 PM

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parmenides51 wrote:

Given a triangle $ABC$ . Let $S$ be the circle passing through $C$ , centered at $A$ . Let $X$ be a variable point on $S$ and let $K$ be the midpoint of the segment $CX$ . Find the locus of the midpoints of $BK$ , when $X$ moves along $S$ .

(I. Gorodnin)

The locus is simple, if $M, N$ are midpoints of segments $AB, BC$ , if $D$ is the foot of perpendicular from $C$ to $AB$ and if $K$ is midpoint of segment $BD$ , then the locus is circumcircle of $\triangle KMN$ . This is just looking at $3$ crucial cases.

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386 posts

#3 h Dec 31, 2020, 12:44 AM • 1 Y

Y by Mango247

$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(12cm); 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 = -38.725305134002674, xmax = 31.077023351644847, ymin = -27.484063950426613, ymax = 16.130985743873076; /* image dimensions */ /* draw figures */ draw(circle((0,0), 8), linewidth(0.8) + red); draw(circle((-5.857055666613099,-5.449302608425723), 15.53315448256311), linewidth(0.8) + red); draw((-5.857055666613099,-5.449302608425723)--(2.6462070486179488,7.549674711922672), linewidth(0.8) + blue); draw((4.5154837090052995,-6.603817598458315)--(2.6462070486179488,7.549674711922672), linewidth(0.8) + blue); draw((-5.857055666613099,-5.449302608425723)--(4.5154837090052995,-6.603817598458315), linewidth(0.8) + blue); draw((2.6462070486179488,7.549674711922672)--(-21.36736632779777,-6.291413867530797), linewidth(0.8) + blue); draw((-9.360579639589911,0.6291304221959377)--(4.5154837090052995,-6.603817598458315), linewidth(0.8) + blue); draw((-5.857055666613099,-5.449302608425723)--(-9.360579639589911,0.6291304221959377), linewidth(0.8) + blue); draw((-21.36736632779777,-6.291413867530797)--(-5.857055666613099,-5.449302608425723), linewidth(0.8) + blue); /* locus construction */ draw((-2.422547965292306,-2.9873435881311887)--(-2.423581329425397,-2.9673597147118898)--(-2.4095598245177823,-3.157444619024452)--(-2.3862281849475306,-3.3466125551086474)--(-2.3536426186779766,-3.534407800837294)--(-2.3118816271924234,-3.720377941014511)--(-2.2610458163771905,-3.9040749572835507)--(-2.201257654153222,-4.085056307442264)--(-2.13266117544014,-4.262885991566023)--(-2.055421635163507,-4.437135602369734)--(-1.9697251101412339,-4.607385357278506)--(-1.8757780508082287,-4.773225109720649)--(-1.773806783859214,-4.93425533720668)--(-1.6640569670078995,-5.090088103814005)--(-1.5467929971760275,-5.240347994758529)--(-1.422297373538035,-5.3846730208017535)--(-1.2908700169557927,-5.522715490314561)--(-1.1528275474429859,-5.654142846896804)--(-1.008502521399762,-5.778638470534798)--(-0.8582426304552393,-5.89590244036667)--(-0.7024098638479148,-6.005652257217986)--(-0.5413796363618837,-6.107623524167001)--(-0.37553988391974213,-6.201570583500008)--(-0.20529012901096966,-6.287267108522281)--(-0.031040518207258394,-6.3645066487989155)--(0.14678916591650104,-6.433103127511998)--(0.32777051607521424,-6.492891289735968)--(0.5114675323442555,-6.543727100551202)--(0.6974376725214735,-6.585488092036756)--(0.8852329182501206,-6.618073658306311)--(1.0744008543343164,-6.641405297876563)--(1.2644857586468792,-6.655426802784179)--(1.4550297000038497,-6.660104393995697)--(1.6455736413608197,-6.65542680278418)--(1.835658545673383,-6.641405297876565)--(2.0248264817575787,-6.618073658306314)--(2.2126217274862263,-6.585488092036761)--(2.3985918676634443,-6.543727100551208)--(2.5822888839324856,-6.492891289735975)--(2.7632702340911997,-6.433103127512007)--(2.94109991821496,-6.364506648798925)--(3.1153495290186712,-6.287267108522292)--(3.285599283927445,-6.201570583500018)--(3.451439036369588,-6.107623524167013)--(3.61246926385562,-6.0056522572179984)--(3.7683020304629458,-5.895902440366683)--(3.9185619214074703,-5.778638470534811)--(4.062886947450695,-5.654142846896817)--(4.200929416963504,-5.5227154903145745)--(4.332356773545746,-5.384673020801767)--(4.4568523971837415,-5.240347994758542)--(4.574116367015614,-5.090088103814018)--(4.683866183866931,-4.934255337206693)--(4.785837450815945,-4.773225109720661)--(4.879784510148952,-4.6073853572785195)--(4.965481035171226,-4.437135602369747)--(5.042720575447861,-4.262885991566035)--(5.111317054160942,-4.085056307442276)--(5.1711052163849125,-3.9040749572835622)--(5.221941027200145,-3.7203779410145215)--(5.263702018685699,-3.534407800837304)--(5.2962875849552535,-3.346612555108657)--(5.319619224525505,-3.1574446190244614)--(5.333640729433121,-2.9673597147118986)--(5.338318320644639,-2.7768157733549286)--(5.333640729433123,-2.5862718319979585)--(5.319619224525507,-2.3961869276853958)--(5.296287584955255,-2.2070189916012)--(5.263702018685702,-2.0192237458725524)--(5.221941027200149,-1.8332536056953348)--(5.171105216384916,-1.6495565894262938)--(5.111317054160947,-1.4685752392675802)--(5.042720575447866,-1.29074555514382)--(4.9654810351712335,-1.1164959443401088)--(4.87978451014896,-0.9462461894313359)--(4.785837450815954,-0.780406436989193)--(4.6838661838669395,-0.619376209503161)--(4.574116367015624,-0.46354344289583516)--(4.456852397183752,-0.3132835519513111)--(4.332356773545758,-0.1689585259080859)--(4.200929416963516,-0.03091605639527728)--(4.062886947450708,0.10051130018696552)--(3.9185619214074836,0.22500692382496013)--(3.7683020304629595,0.34227089365683305)--(3.612469263855634,0.45202071050814885)--(3.4514390363696026,0.5539919774571644)--(3.28559928392746,0.647939036790171)--(3.115349529018687,0.733635561812445)--(2.9410999182149755,0.8108751020890788)--(2.7632702340912156,0.8794715808021607)--(2.582288883932502,0.9392597430261302)--(2.398591867663461,0.990095553841364)--(2.212621727486243,1.0318565453269182)--(2.024826481757596,1.0644421115964722)--(1.8356585456734003,1.0877737511667238)--(1.645573641360837,1.1017952560743396)--(1.4550297000038666,1.1064728472858576)--(1.2644857586468965,1.1017952560743396)--(1.0744008543343333,1.0877737511667247)--(0.885232918250138,1.064442111596473)--(0.6974376725214904,1.03185654532692)--(0.5114675323442723,0.9900955538413658)--(0.3277705160752311,0.9392597430261329)--(0.14678916591651747,0.8794715808021643)--(-0.03104051820724285,0.8108751020890823)--(-0.20529012901095411,0.7336355618124486)--(-0.37553988391972704,0.6479390367901754)--(-0.5413796363618686,0.5539919774571693)--(-0.7024098638478997,0.45202071050815507)--(-0.8582426304552251,0.34227089365684016)--(-1.0085025213997487,0.22500692382496856)--(-1.1528275474429726,0.10051130018697485)--(-1.2908700169557812,-0.030916056395266622)--(-1.4222973735380235,-0.16895852590807436)--(-1.5467929971760168,-0.3132835519512982)--(-1.6640569670078897,-0.46354344289582095)--(-1.773806783859205,-0.6193762095031459)--(-1.8757780508082198,-0.7804064369891766)--(-1.9697251101412268,-0.9462461894313181)--(-2.0554216351635,-1.1164959443400906)--(-2.1326611754401337,-1.290745555143801)--(-2.201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linewidth(0.8) + red); /* locus construction */ draw((-9.360579639589911,0.6291304221959377)--(-9.362646367856094,0.6690981690345352)--(-9.350957672785578,0.47884118633048933)--(-9.334603358040864,0.2889283604094106)--(-9.313593274849627,0.09947408765633625)--(-9.28794007890036,-0.08940751175897921)--(-9.257659222719074,-0.2776026626237855)--(-9.222768946361253,-0.4649980032162726)--(-9.183290266424747,-0.6514806535904296)--(-9.139246963390146,-0.8369382835707069)--(-9.090665567296314,-1.0212591804155111)--(-9.03757534175968,-1.2043323161087867)--(-8.980008266346937,-1.3860474142391372)--(-8.917999017311743,-1.566295016426213)--(-8.85158494670704,-1.7449665482543497)--(-8.78080605988558,-1.921954384673731)--(-8.70570499140219,-2.0971519148297024)--(-8.626326979332314,-2.2704536062811527)--(-8.54271983802229,-2.441755068569321)--(-8.454933929287767,-2.6109531160986976)--(-8.363022132077646,-2.777945830292176)--(-8.267039810621757,-2.9426326209829825)--(-8.167044781081538,-3.104914287006443)--(-8.063097276723727,-3.2646930759550448)--(-7.955259911638105,-3.421872743060842)--(-7.8435976430210985,-3.5763586091696955)--(-7.728177732048001,-3.728057617772459)--(-7.609069703357354,-3.8768783910587428)--(-7.486345303171907,-4.022731284959484)--(-7.36007845608137,-4.165528443145192)--(-7.230345220513005,-4.3051838499473)--(-7.097223742916885,-4.441613382170807)--(-6.960794210693379,-4.574734859766927)--(-6.821138803891271,-4.704468095335292)--(-6.678341645705565,-4.830734942425831)--(-6.532488751804824,-4.95345934261128)--(-6.38366797851854,-5.072567371301927)--(-6.231968969915778,-5.187987282275024)--(-6.077483103806925,-5.299649550892033)--(-5.920303436701129,-5.407486915977657)--(-5.760524647752526,-5.511434420335467)--(-5.598242981729067,-5.611429449875686)--(-5.43355619103826,-5.707411771331575)--(-5.266563476844784,-5.7993235685417)--(-5.097365429315406,-5.887109477276221)--(-4.926063967027239,-5.970716618586247)--(-4.752762275575789,-6.050094630656124)--(-4.577564745419818,-6.125195699139516)--(-4.400576909000437,-6.195974585960975)--(-4.2219053771723,-6.26238865656568)--(-4.041657774985225,-6.3243979056008754)--(-3.8599426768548746,-6.381964981013619)--(-3.6768695411616,-6.435055206550253)--(-3.492548644316795,-6.483636602644086)--(-3.3070910143365184,-6.5276799056786885)--(-3.1206083639623614,-6.567158585615195)--(-2.933213023369875,-6.6020488619730155)--(-2.745017872505068,-6.632329718154304)--(-2.556136273089753,-6.65798291410357)--(-2.366682000336679,-6.678992997294809)--(-2.1767691744156004,-6.695347312039525)--(-1.9865121917115547,-6.707036007110043)--(-1.7960256559167556,-6.7140520416735185)--(-1.605424308997616,-6.716391189533079)--(-1.4148229620784762,-6.71405204167352)--(-1.2243364262836771,-6.707036007110046)--(-1.0340794435796312,-6.695347312039532)--(-0.8441666176585525,-6.678992997294818)--(-0.6547123449054781,-6.657982914103581)--(-0.46583074549016246,-6.632329718154315)--(-0.2776355946253557,-6.60204886197303)--(-0.09024025403286862,-6.567158585615212)--(0.0962423963412884,-6.527679905678706)--(0.2817000263215659,-6.483636602644108)--(0.46602092316637145,-6.435055206550274)--(0.649094058859647,-6.381964981013642)--(0.8308091569899971,-6.3243979056009)--(1.0110567591770738,-6.262388656565706)--(1.18972829100521,-6.195974585961006)--(1.3667161274245927,-6.125195699139546)--(1.541913657580564,-6.050094630656156)--(1.7152153490320152,-5.970716618586282)--(1.8865168113201836,-5.887109477276258)--(2.0557148588495613,-5.7993235685417375)--(2.2227075730430395,-5.707411771331616)--(2.387394363733847,-5.611429449875729)--(2.5496760297573067,-5.51143442033551)--(2.709454818705911,-5.407486915977701)--(2.8666344858117094,-5.299649550892079)--(3.0211203519205636,-5.187987282275072)--(3.172819360523329,-5.072567371301975)--(3.3216401338096135,-4.953459342611328)--(3.467493027710357,-4.830734942425881)--(3.6102901858960648,-4.704468095335342)--(3.7499455926981744,-4.574734859766979)--(3.886375124921682,-4.441613382170859)--(4.019496602517805,-4.305183849947353)--(4.14922983808617,-4.165528443145245)--(4.27549668517671,-4.022731284959539)--(4.39822108536216,-3.876878391058798)--(4.517329114052809,-3.728057617772514)--(4.632749025025908,-3.5763586091697506)--(4.744411293642917,-3.4218727430608973)--(4.8522486587285405,-3.2646930759551016)--(4.956196163086353,-3.104914287006498)--(5.0561911926265735,-2.9426326209830376)--(5.152173514082464,-2.7779458302922317)--(5.244085311292588,-2.6109531160987545)--(5.331871220027113,-2.441755068569377)--(5.415478361337138,-2.2704536062812086)--(5.494856373407016,-2.0971519148297575)--(5.569957441890407,-1.9219543846737879)--(5.640736328711869,-1.7449665482544048)--(5.707150399316573,-1.5662950164262681)--(5.769159648351771,-1.3860474142391923)--(5.826726723764514,-1.2043323161088417)--(5.87981694930115,-1.0212591804155653)--(5.928398345394983,-0.8369382835707602)--(5.9724416484295855,-0.6514806535904829)--(6.011920328366093,-0.464998003216325)--(6.046810604723914,-0.2776026626238388)--(6.077091460905203,-0.0894075117590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7)--(-9.222768946361233,2.5653701067133206)--(-9.257659222719056,2.3779747661208344)--(-9.287940078900345,2.189779615256028)--(-9.313593274849612,2.000898015840713)--(-9.334603358040853,1.8114437430876387)--(-9.35095767278557,1.62153091716656)--(-9.362646367856089,1.4312739344625145)--(-9.369662402419568,1.2407873986677154)--(-9.37200155027913,1.0501860517485757)^^(-9.360579639589911,0.6291304221959377), linewidth(0.8) + red); draw((2.6462070486179488,7.549674711922672)--(1.1130058672144145,-6.225104967987619), linewidth(0.8) + blue); /* dots and labels */ dot((-5.857055666613099,-5.449302608425723),dotstyle); label("$A$", (-5.694660804846591,-4.992202549495916), NE * labelscalefactor); dot((4.5154837090052995,-6.603817598458315),dotstyle); label("$B$", (4.707254812622452,-6.132763472464004), NE * labelscalefactor); dot((2.6462070486179488,7.549674711922672),dotstyle); label("$C$", (2.882357335873497,7.78207978774667), NE * labelscalefactor); dot((-21.36736632779777,-6.291413867530797),dotstyle); label("$X$", (-21.206289357212707,-5.813406414032939), NE * labelscalefactor); dot((-9.360579639589911,0.6291304221959377),linewidth(4pt) + dotstyle); label("$K$", (-9.161966010669605,0.9843366868568656), NE * labelscalefactor); dot((-2.422547965292306,-2.9873435881311887),linewidth(4pt) + dotstyle); label("$M$", (-3.550406269666569,-3.532284568096763), NE * labelscalefactor); dot((-0.6707859788038997,-6.026560103442019),linewidth(4pt) + dotstyle); label("$N$", (-0.49370299611206936,-5.6765391032767685), NE * labelscalefactor); dot((3.580845378811624,0.4729285567321786),linewidth(4pt) + dotstyle); label("$G$", (3.7491836373292506,0.847469376100695), NE * labelscalefactor); dot((1.1130058672144145,-6.225104967987619),linewidth(4pt) + dotstyle); label("$D$", (1.2855720437181615,-5.859028850951663), NE * labelscalefactor); dot((2.8142447881098582,-6.414461283222967),linewidth(4pt) + dotstyle); label("$P$", (3.019224646629669,-6.041518598626557), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
Nice problem, gets easily obliterated by homothethy

$\color{black}\rule{25cm}{1pt}$
Since we have that $X$ moves along $S$ , then so must $K$ .
This stems from the fact that $\angle AKC= 90$ for any $X$ we choose on the circle.
We denote this circle around $KAC$ with $\Gamma$ .
Let $D$ be the foot of the C-altitude of $ABC$ .
We easily that $D$ must be on $\Gamma$ , since we have that $\angle ADC = 90$ .
Now let's do a homothety centered at $B$ with coefficient $\frac{1}{2}$ and let's call it $\mathcal{H}$ .
By this way we have that $K \overset{\mathcal{H}}{\rightarrow} M$ and $C$ and $A$ get sent to the midpoints of $CB$ and $AB$ , respectively.(we call them $G$ and $N$ , respectively) and $D$ gets sent to a point $P$ such that $\frac{BD}{BP}=2$ and all of those points are on a circle with radius half of what $\Gamma$ has, we call this new circle $\omega$
But notice that $\Gamma$ has a radius of $\frac{1}{2}AC$ which is fixed so that means that $\omega$ has a radius of $\frac{1}{4}AC$ and since the points $N,G,P$ are all fixed that means that $M$ moves along a fixed circle.

Z K Y

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126 posts

#4 h Jan 5, 2021, 12:57 PM

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EulersTurban wrote:

$[asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(12cm); 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 = -38.725305134002674, xmax = 31.077023351644847, ymin = -27.484063950426613, ymax = 16.130985743873076; /* image dimensions */ /* draw figures */ draw(circle((0,0), 8), linewidth(0.8) + red); draw(circle((-5.857055666613099,-5.449302608425723), 15.53315448256311), linewidth(0.8) + red); draw((-5.857055666613099,-5.449302608425723)--(2.6462070486179488,7.549674711922672), linewidth(0.8) + blue); draw((4.5154837090052995,-6.603817598458315)--(2.6462070486179488,7.549674711922672), linewidth(0.8) + blue); draw((-5.857055666613099,-5.449302608425723)--(4.5154837090052995,-6.603817598458315), linewidth(0.8) + blue); draw((2.6462070486179488,7.549674711922672)--(-21.36736632779777,-6.291413867530797), linewidth(0.8) + blue); draw((-9.360579639589911,0.6291304221959377)--(4.5154837090052995,-6.603817598458315), linewidth(0.8) + blue); draw((-5.857055666613099,-5.449302608425723)--(-9.360579639589911,0.6291304221959377), linewidth(0.8) + blue); draw((-21.36736632779777,-6.291413867530797)--(-5.857055666613099,-5.449302608425723), linewidth(0.8) + blue); /* locus construction */ draw((-2.422547965292306,-2.9873435881311887)--(-2.423581329425397,-2.9673597147118898)--(-2.4095598245177823,-3.157444619024452)--(-2.3862281849475306,-3.3466125551086474)--(-2.3536426186779766,-3.534407800837294)--(-2.3118816271924234,-3.720377941014511)--(-2.2610458163771905,-3.9040749572835507)--(-2.201257654153222,-4.085056307442264)--(-2.13266117544014,-4.262885991566023)--(-2.055421635163507,-4.437135602369734)--(-1.9697251101412339,-4.607385357278506)--(-1.8757780508082287,-4.773225109720649)--(-1.773806783859214,-4.93425533720668)--(-1.6640569670078995,-5.090088103814005)--(-1.5467929971760275,-5.240347994758529)--(-1.422297373538035,-5.3846730208017535)--(-1.2908700169557927,-5.522715490314561)--(-1.1528275474429859,-5.654142846896804)--(-1.008502521399762,-5.778638470534798)--(-0.8582426304552393,-5.89590244036667)--(-0.7024098638479148,-6.005652257217986)--(-0.5413796363618837,-6.107623524167001)--(-0.37553988391974213,-6.201570583500008)--(-0.20529012901096966,-6.287267108522281)--(-0.031040518207258394,-6.3645066487989155)--(0.14678916591650104,-6.433103127511998)--(0.32777051607521424,-6.492891289735968)--(0.5114675323442555,-6.543727100551202)--(0.6974376725214735,-6.585488092036756)--(0.8852329182501206,-6.618073658306311)--(1.0744008543343164,-6.641405297876563)--(1.2644857586468792,-6.655426802784179)--(1.4550297000038497,-6.660104393995697)--(1.6455736413608197,-6.65542680278418)--(1.835658545673383,-6.641405297876565)--(2.0248264817575787,-6.618073658306314)--(2.2126217274862263,-6.585488092036761)--(2.3985918676634443,-6.543727100551208)--(2.5822888839324856,-6.492891289735975)--(2.7632702340911997,-6.433103127512007)--(2.94109991821496,-6.364506648798925)--(3.1153495290186712,-6.287267108522292)--(3.285599283927445,-6.201570583500018)--(3.451439036369588,-6.107623524167013)--(3.61246926385562,-6.0056522572179984)--(3.7683020304629458,-5.895902440366683)--(3.9185619214074703,-5.778638470534811)--(4.062886947450695,-5.654142846896817)--(4.200929416963504,-5.5227154903145745)--(4.332356773545746,-5.384673020801767)--(4.4568523971837415,-5.240347994758542)--(4.574116367015614,-5.090088103814018)--(4.683866183866931,-4.934255337206693)--(4.785837450815945,-4.773225109720661)--(4.879784510148952,-4.6073853572785195)--(4.965481035171226,-4.437135602369747)--(5.042720575447861,-4.262885991566035)--(5.111317054160942,-4.085056307442276)--(5.1711052163849125,-3.9040749572835622)--(5.221941027200145,-3.7203779410145215)--(5.263702018685699,-3.534407800837304)--(5.2962875849552535,-3.346612555108657)--(5.319619224525505,-3.1574446190244614)--(5.333640729433121,-2.9673597147118986)--(5.338318320644639,-2.7768157733549286)--(5.333640729433123,-2.5862718319979585)--(5.319619224525507,-2.3961869276853958)--(5.296287584955255,-2.2070189916012)--(5.263702018685702,-2.0192237458725524)--(5.221941027200149,-1.8332536056953348)--(5.171105216384916,-1.6495565894262938)--(5.111317054160947,-1.4685752392675802)--(5.042720575447866,-1.29074555514382)--(4.9654810351712335,-1.1164959443401088)--(4.87978451014896,-0.9462461894313359)--(4.785837450815954,-0.780406436989193)--(4.6838661838669395,-0.619376209503161)--(4.574116367015624,-0.46354344289583516)--(4.456852397183752,-0.3132835519513111)--(4.332356773545758,-0.1689585259080859)--(4.200929416963516,-0.03091605639527728)--(4.062886947450708,0.10051130018696552)--(3.9185619214074836,0.22500692382496013)--(3.7683020304629595,0.34227089365683305)--(3.612469263855634,0.45202071050814885)--(3.4514390363696026,0.5539919774571644)--(3.28559928392746,0.647939036790171)--(3.115349529018687,0.733635561812445)--(2.9410999182149755,0.8108751020890788)--(2.7632702340912156,0.8794715808021607)--(2.582288883932502,0.9392597430261302)--(2.398591867663461,0.990095553841364)--(2.212621727486243,1.0318565453269182)--(2.024826481757596,1.0644421115964722)--(1.8356585456734003,1.0877737511667238)--(1.645573641360837,1.1017952560743396)--(1.4550297000038666,1.1064728472858576)--(1.2644857586468965,1.1017952560743396)--(1.0744008543343333,1.0877737511667247)--(0.885232918250138,1.064442111596473)--(0.6974376725214904,1.03185654532692)--(0.5114675323442723,0.9900955538413658)--(0.3277705160752311,0.9392597430261329)--(0.14678916591651747,0.8794715808021643)--(-0.03104051820724285,0.8108751020890823)--(-0.20529012901095411,0.7336355618124486)--(-0.37553988391972704,0.6479390367901754)--(-0.5413796363618686,0.5539919774571693)--(-0.7024098638478997,0.45202071050815507)--(-0.8582426304552251,0.34227089365684016)--(-1.0085025213997487,0.22500692382496856)--(-1.1528275474429726,0.10051130018697485)--(-1.2908700169557812,-0.030916056395266622)--(-1.4222973735380235,-0.16895852590807436)--(-1.5467929971760168,-0.3132835519512982)--(-1.6640569670078897,-0.46354344289582095)--(-1.773806783859205,-0.6193762095031459)--(-1.8757780508082198,-0.7804064369891766)--(-1.9697251101412268,-0.9462461894313181)--(-2.0554216351635,-1.1164959443400906)--(-2.1326611754401337,-1.290745555143801)--(-2.201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linewidth(0.8) + red); /* locus construction */ 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linewidth(0.8) + red); draw((2.6462070486179488,7.549674711922672)--(1.1130058672144145,-6.225104967987619), linewidth(0.8) + blue); /* dots and labels */ dot((-5.857055666613099,-5.449302608425723),dotstyle); label("$A$", (-5.694660804846591,-4.992202549495916), NE * labelscalefactor); dot((4.5154837090052995,-6.603817598458315),dotstyle); label("$B$", (4.707254812622452,-6.132763472464004), NE * labelscalefactor); dot((2.6462070486179488,7.549674711922672),dotstyle); label("$C$", (2.882357335873497,7.78207978774667), NE * labelscalefactor); dot((-21.36736632779777,-6.291413867530797),dotstyle); label("$X$", (-21.206289357212707,-5.813406414032939), NE * labelscalefactor); dot((-9.360579639589911,0.6291304221959377),linewidth(4pt) + dotstyle); label("$K$", (-9.161966010669605,0.9843366868568656), NE * labelscalefactor); dot((-2.422547965292306,-2.9873435881311887),linewidth(4pt) + dotstyle); label("$M$", (-3.550406269666569,-3.532284568096763), NE * labelscalefactor); dot((-0.6707859788038997,-6.026560103442019),linewidth(4pt) + dotstyle); label("$N$", (-0.49370299611206936,-5.6765391032767685), NE * labelscalefactor); dot((3.580845378811624,0.4729285567321786),linewidth(4pt) + dotstyle); label("$G$", (3.7491836373292506,0.847469376100695), NE * labelscalefactor); dot((1.1130058672144145,-6.225104967987619),linewidth(4pt) + dotstyle); label("$D$", (1.2855720437181615,-5.859028850951663), NE * labelscalefactor); dot((2.8142447881098582,-6.414461283222967),linewidth(4pt) + dotstyle); label("$P$", (3.019224646629669,-6.041518598626557), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy]$
Nice problem, gets easily obliterated by homothethy

$\color{black}\rule{25cm}{1pt}$
Since we have that $X$ moves along $S$ , then so must $K$ .
This stems from the fact that $\angle AKC= 90$ for any $X$ we choose on the circle.
We denote this circle around $KAC$ with $\Gamma$ .
Let $D$ be the foot of the C-altitude of $ABC$ .
We easily that $D$ must be on $\Gamma$ , since we have that $\angle ADC = 90$ .
Now let's do a homothety centered at $B$ with coefficient $\frac{1}{2}$ and let's call it $\mathcal{H}$ .
By this way we have that $K \overset{\mathcal{H}}{\rightarrow} M$ and $C$ and $A$ get sent to the midpoints of $CB$ and $AB$ , respectively.(we call them $G$ and $N$ , respectively) and $D$ gets sent to a point $P$ such that $\frac{BD}{BP}=2$ and all of those points are on a circle with radius half of what $\Gamma$ has, we call this new circle $\omega$
But notice that $\Gamma$ has a radius of $\frac{1}{2}AC$ which is fixed so that means that $\omega$ has a radius of $\frac{1}{4}AC$ and since the points $N,G,P$ are all fixed that means that $M$ moves along a fixed circle.

Homothety takes a circle to circle just check 3 cases that's it as 3 points uniquely define a circle

Z K Y

N Quick Reply

G

H

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