Let ABC be a triangle with M,N,P as midpoints of the segments BC,CA,AB respectively. Suppose that I is the intersection of angle bisectors of ∠BPM,∠MNP and J is the intersection of angle bisectors of ∠CNM,∠MPN. Denote (ω1) as the circle of center I and tangent to MP at D, (ω2) as the circle of center J and tangent to MN at E. a) Prove that DE is parallel to BC. b) Prove that the radical axis of two circles (ω1),(ω2) bisects the segment DE.
Problem
Source: 2018 Saudi Arabia BMO TST II p1
Tags: radical axis, geometry, bisects segment
18.10.2021 06:17
Solved with sotpidot a) Note IP∥JN by angle chasing. If we let Q=PC∩IN, then MQ∥IP due to Q being the incenter of △MNP. By angle bisector theorem, and by △IPD∼△JNE, we have IDJE=ACABAdditionally, we see IPJN=MPMN⟹MPMN=PDNEimplying DE∥BC∥NP, as desired. ◼. b) Let S1=DE∩ω1 and S2=DE∩ω2. Then, DS2ES1=PDsin∠PDS1NEsin∠NES2=ACsin∠ACBABsin∠ABC=1hence DS1=ES2. If L is the midpoint of DE, then LD⋅LS1=LE⋅LS2which implies L lies on the radical axis, as desired. ◼ [asy][asy] import graph; size(16cm); 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 = 1.6521126872149106, xmax = 27.264956864035277, ymin = -7.498208401550829, ymax = 9.547273263162316; /* image dimensions */ pen ffqqff = rgb(1,0,1); pen wwzzqq = rgb(0.4,0.6,0); /* draw figures */ draw((4.44,-1.12)--(7.6,7.02), linewidth(1)); draw((7.6,7.02)--(17.76,-0.94), linewidth(1)); draw((17.76,-0.94)--(4.44,-1.12), linewidth(1) + blue); draw((xmin, -3.6421405488184306*xmin + 24.875686103886952)--(xmax, -3.6421405488184306*xmax + 24.875686103886952), linewidth(1) + dotted); /* line */ draw((xmin, 0.694470665794304*xmin-5.765888042271775)--(xmax, 0.694470665794304*xmax-5.765888042271775), linewidth(1) + dotted); /* line */ draw((xmin, -3.6421405488184293*xmin + 49.22234215901768)--(xmax, -3.6421405488184293*xmax + 49.22234215901768), linewidth(1) + dotted); /* line */ draw((xmin, -0.33754028021171556*xmin + 4.981992486874527)--(xmax, -0.33754028021171556*xmax + 4.981992486874527), linewidth(1) + dotted); /* line */ draw((6.02,2.95)--(11.1,-1.03), linewidth(1)); draw(circle((7.065787692221115,-0.8589057592937778), 2.3533219890404102), linewidth(1) + ffqqff); draw((11.1,-1.03)--(12.68,3.04), linewidth(1)); draw(circle((13.387504108264137,0.46317059883555706), 1.592086986392247), linewidth(1) + ffqqff); draw((8.5171425344622,0.9935773056772542)--(11.903329684163333,1.0393365914840254), linewidth(1) + blue); draw((4.44,-1.12)--(3.6095689751522393,-3.2591482728673355), linewidth(1)); draw((5.564904863138467,0.9536822020107177)--(8.5171425344622,0.9935773056772542), linewidth(1) + dotted + wwzzqq); draw((11.903329684163333,1.0393365914840254)--(14.855567355487052,1.079231695150561), linewidth(1) + dotted + wwzzqq); /* dots and labels */ dot((4.44,-1.12),dotstyle); label("B", (3.8385749949922587,-0.8272672788424127), NE * labelscalefactor); dot((7.6,7.02),dotstyle); label("A", (7.698350293415538,7.249256756008579), NE * labelscalefactor); dot((17.76,-0.94),dotstyle); label("C", (17.849782436667514,-0.7157130794660177), NE * labelscalefactor); dot((11.1,-1.03),linewidth(4pt) + dotstyle); label("M", (11.178841313959072,-0.8495781187176916), NE * labelscalefactor); dot((12.68,3.04),linewidth(4pt) + dotstyle); label("N", (12.762910945103886,3.2109947385830835), NE * labelscalefactor); dot((6.02,2.95),linewidth(4pt) + dotstyle); label("P", (6.3150782211482355,3.0994405392066886), NE * labelscalefactor); dot((7.065787692221115,-0.8589057592937778),linewidth(4pt) + dotstyle); label("I", (7.16289013640884,-0.6710913997154597), NE * labelscalefactor); dot((13.387504108264137,0.46317059883555706),linewidth(4pt) + dotstyle); label("J", (13.476857821112818,0.6452481529260002), NE * labelscalefactor); dot((8.5171425344622,0.9935773056772542),linewidth(4pt) + dotstyle); label("D", (8.501540528925585,1.0914649504315799), NE * labelscalefactor); label("ω1", (4.976427828631492,-0.5595372003390648), NE * labelscalefactor,ffqqff); dot((11.903329684163333,1.0393365914840254),linewidth(4pt) + dotstyle); label("E", (11.379638872836583,1.3591950289349277), NE * labelscalefactor); label("ω2", (13.566101180613932,1.2030191498079748), NE * labelscalefactor,ffqqff); dot((10.210236109312767,1.0164569485806398),linewidth(4pt) + dotstyle); label("L", (10.308718558823188,0.1767205155451416), NE * labelscalefactor); dot((10.414502453429996,1.4666784104781674),linewidth(4pt) + dotstyle); label("Q", (10.152542679696234,1.8723443460663443), NE * labelscalefactor); dot((3.6095689751522393,-3.2591482728673355),linewidth(4pt) + dotstyle); label("R", (2.901519720230538,-3.102972946120869), NE * labelscalefactor); dot((5.564904863138467,0.9536822020107177),linewidth(4pt) + dotstyle); label("S1", (5.288779586885398,0.042855476293467705), NE * labelscalefactor); dot((14.855567355487052,1.079231695150561),linewidth(4pt) + dotstyle); label("S2", (15.038616612382352,0.19903135542042058), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy][/asy]