We prove $A_1I_A'$ pass through $I$ the incenter of $\triangle ABC$. Let $X=AA_1 \cap I_aI_a'$,$D=AI_a \cap BC$ and $I'=A_1I_a' \cap AI_a$. Note $I_aI_a' \parallel BC$ $$(I_a',I_a;X,\infty{\overline{I_aI_a'}})\stackrel{A_1}{=}(I',I_a;A,D)=-1=(I,I_a;A,D)$$ hence $I'=I$. Doing same for all three vertex give us $I$ is commmon point. [asy][asy] /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */ import graph; size(13.25506243475847cm); 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 = -34.379252619388744, xmax = 48.87580981536973, ymin = -26.22217227151488, ymax = 30.350219922211807; /* image dimensions */ pen ffxfqq = rgb(1.,0.4980392156862745,0.); pen qqwuqq = rgb(0.,0.39215686274509803,0.); /* draw figures */ draw(circle((11.466171703141281,0.04284135030464683), 10.555283422054734), linewidth(0.4)); draw((5.551799596175703,8.785507459693191)--(14.289330248990318,-20.956069749899253), linewidth(0.4) + red); draw((5.551799596175703,8.785507459693191)--(5.236404683936008,-4.753084292660743), linewidth(0.4)); draw((5.551799596175703,8.785507459693191)--(-4.5612198355518725,-20.51692750391562), linewidth(0.4)); draw((-4.5612198355518725,-20.51692750391562)--(14.289330248990318,-20.956069749899253), linewidth(0.4)); draw((5.236404683936008,-4.753084292660743)--(4.864055206719224,-20.736498626907437), linewidth(0.4)); draw(circle((14.780988431030714,0.14877103868872404), 21.110566844109467), linewidth(0.4)); draw((8.15135497525186,-0.06308833807942849)--(-4.5612198355518725,-20.51692750391562), linewidth(0.4) + red); draw((5.236404683936008,-4.753084292660743)--(14.289330248990318,-20.956069749899253), linewidth(0.4)); draw((20.042978514568922,23.98335355016288)--(8.15135497525186,-0.06308833807942849), linewidth(0.4)); draw((-7.463023899641593,-1.4602998693509348)--(8.15135497525186,-0.06308833807942849), linewidth(0.4)); draw((5.551799596175703,8.785507459693191)--(2.02556912852416,-4.678284696311357), linewidth(0.4) + ffxfqq); draw((2.02556912852416,-4.678284696311357)--(20.676685636948726,-5.112780939491454), linewidth(0.4) + ffxfqq); draw((20.676685636948726,-5.112780939491454)--(5.551799596175703,8.785507459693191), linewidth(0.4) + ffxfqq); draw((-5.64078785537825,5.4973302640763)--(29.06478944370122,15.693193225120972), linewidth(0.4) + qqwuqq); draw((29.06478944370122,15.693193225120972)--(14.289330248990318,-20.956069749899253), linewidth(0.4) + qqwuqq); draw((14.289330248990318,-20.956069749899253)--(-5.64078785537825,5.4973302640763), linewidth(0.4) + qqwuqq); /* dots and labels */ dot((5.551799596175703,8.785507459693191),dotstyle); label("$A$", (5.837175844858993,9.44024276217224), NE * labelscalefactor); dot((2.02556912852416,-4.678284696311357),dotstyle); label("$B$", (0.8341751746654944,-5.37633614570856), NE * labelscalefactor); dot((20.676685636948726,-5.112780939491454),dotstyle); label("$C$", (21.231024060838987,-5.504618214175061), NE * labelscalefactor); dot((5.321710955366024,-1.091240876718917),linewidth(4.pt) + dotstyle); label("$H$", (5.7730348106257425,-0.5016175439815439), NE * labelscalefactor); dot((5.236404683936008,-4.753084292660743),linewidth(4.pt) + dotstyle); label("$A_1$", (3.8488037836282434,-3.9652333925770553), NE * labelscalefactor); dot((10.347812856674986,4.378441166364202),linewidth(4.pt) + dotstyle); label("$B'$", (10.006343070020241,5.463498639710726), NE * labelscalefactor); dot((3.1163101903918906,-0.5136360473569644),linewidth(4.pt) + dotstyle); label("$C'$", (3.5280986124619935,-0.24505340704854303), NE * labelscalefactor); dot((8.15135497525186,-0.06308833807942849),linewidth(4.pt) + dotstyle); label("$I$", (9.108368590754742,-0.24505340704854303), NE * labelscalefactor); dot((29.06478944370122,15.693193225120972),linewidth(4.pt) + dotstyle); label("$I_B$", (29.31279437422848,16.17505135666351), NE * labelscalefactor); dot((-5.64078785537825,5.4973302640763),linewidth(4.pt) + dotstyle); label("$I_C$", (-7.055172036024253,6.81046035860898), NE * labelscalefactor); dot((14.289330248990318,-20.956069749899253),linewidth(4.pt) + dotstyle); label("$I_A$", (15.330048911379988,-20.32119712205586), NE * labelscalefactor); dot((-4.5612198355518725,-20.51692750391562),linewidth(4.pt) + dotstyle); label("$I_A'$", (-5.323364111726503,-19.551504711256857), NE * labelscalefactor); dot((4.864055206719224,-20.736498626907437),linewidth(4.pt) + dotstyle); label("$X$", (5.131624468293243,-20.19291505358936), NE * labelscalefactor); dot((9.5587718894924,-4.853778120379358),linewidth(4.pt) + dotstyle); label("$D$", (9.81391996732049,-4.350079597976556), NE * labelscalefactor); dot((11.466171703141281,0.04284135030464853),linewidth(4.pt) + dotstyle); label("$O$", (11.73815099431799,0.5246390037504595), NE * labelscalefactor); dot((14.780988431030714,0.1487710386887222),linewidth(4.pt) + dotstyle); label("$V$", (15.009343740213739,0.6529210722169599), NE * labelscalefactor); dot((11.220342612121089,-10.509579043989339),linewidth(4.pt) + dotstyle); label("$L$", (11.48158685738499,-9.994490610502575), NE * labelscalefactor); dot((-7.463023899641593,-1.4602998693509348),linewidth(4.pt) + dotstyle); label("$I_C'$", (-7.183454104490752,-0.9506047836142953), NE * labelscalefactor); dot((20.042978514568922,23.98335355016288),linewidth(4.pt) + dotstyle); label("$I_B'$", (20.268908547340235,24.513385806986037), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy][/asy]

an alternate approach