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 $\angle BPM, \angle MNP$ and $J$ is the intersection of angle bisectors of $\angle CN M, \angle MPN$. Denote $(\omega_1)$ as the circle of center $I$ and tangent to $MP$ at $D$, $(\omega_2)$ 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 $(\omega_1), (\omega_2)$ bisects the segment $DE$.
Problem
Source: 2018 Saudi Arabia BMO TST II p1
Tags: radical axis, geometry, bisects segment
JustinLee2017
18.10.2021 06:17
Solved with sotpidot a) Note $IP \parallel JN$ by angle chasing. If we let $Q = PC \cap IN,$ then $MQ \parallel IP$ due to $Q$ being the incenter of $\triangle MNP$. By angle bisector theorem, and by $\triangle IPD \sim \triangle JNE$, we have $$\frac{ID}{JE} = \frac{AC}{AB}$$Additionally, we see $$\frac{IP}{JN} = \frac{MP}{MN} \implies \frac{MP}{MN} = \frac{PD}{NE}$$implying $DE \parallel BC \parallel NP$, as desired. $\quad \blacksquare$. b) Let $S_1 = DE \cap \omega_1$ and $S_2 = DE \cap \omega_2$. Then, $$\frac{DS_2}{ES_1} = \frac{PD \sin \angle PDS_1}{NE \sin \angle NES_2} = \frac{AC \sin \angle ACB}{AB \sin \angle ABC} = 1$$hence $DS_1 = ES_2$. If $L$ is the midpoint of $DE$, then $$LD \cdot LS_1 = LE \cdot LS_2$$which implies $L$ lies on the radical axis, as desired. $\quad \blacksquare$ [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("$\omega_1$", (4.976427828631492,-0.5595372003390648), NE * labelscalefactor,ffqqff); dot((11.903329684163333,1.0393365914840254),linewidth(4pt) + dotstyle); label("$E$", (11.379638872836583,1.3591950289349277), NE * labelscalefactor); label("$\omega_2$", (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("$S_1$", (5.288779586885398,0.042855476293467705), NE * labelscalefactor); dot((14.855567355487052,1.079231695150561),linewidth(4pt) + dotstyle); label("$S_2$", (15.038616612382352,0.19903135542042058), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy][/asy]