It's well known that reflecting the orthocenter over the sides of a triangle leads the resulting points to lie on the circumcircle. Hence, reflecting $H$ over the points $P$ and $Q$, where $P$ and $Q$ are the points symmetric to $O$ wrt $BA$ and $BC$ respectively, gives $HP = HQ = BP = BQ = R$ where $R$ is the circumradius. From this, we see $PA_0C_0Q$ are collinear. Additionally, note that $BOCQ$ and $BOAP$ are rhombi, implying $AP \parallel OB \parallel CQ$. Hence, $ACPQ$ is a parallelogram, and $PQ = AC$. From earlier, we can easily see that $PA_0 = A_0O$ and $QC_0 = C_0O$, so the perimeter of $A_0OC_0$ is equal to $AC$. $\quad \blacksquare$ [asy][asy] import graph; size(7cm); 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 = -8.322024246002622, xmax = 13.310301693771057, ymin = -7.858026352116552, ymax = 14.003213089665309; /* image dimensions */ pen qqwwzz = rgb(0,0.4,0.6); pen qqttzz = rgb(0,0.2,0.6); /* draw figures */ draw((-3.5720690793327403,-1.1909206061281046)--(7.083365948870123,2.620144128539957), linewidth(1)); draw((1.9504691566147727,9.396328861750545)--(-3.5720690793327403,-1.1909206061281046), linewidth(1)); draw((1.9504691566147727,9.396328861750545)--(7.083365948870123,2.620144128539957), linewidth(1) + qqwwzz); draw((-3.5720690793327403,-1.1909206061281046)--(3.7526748149394593,4.357504638941627), linewidth(1)); draw((-1.2215881741533368,3.315182683594067)--(0.8545456056063478,3.2340238726103854), linewidth(1)); draw((0.8545456056063478,3.2340238726103854)--(0.9630550791826599,0.43112970923797356), linewidth(1)); draw((0.9630550791826599,0.43112970923797356)--(-1.2215881741533368,3.315182683594067), linewidth(1) + qqwwzz); draw((-3.5720690793327403,-1.1909206061281046)--(-2.4761455283243157,4.971384383012055), linewidth(1) + qqttzz); draw((-3.5720690793327403,-1.1909206061281046)--(2.656751263931035,-1.8048003501985326), linewidth(1) + qqttzz); draw((-2.4761455283243157,4.971384383012055)--(-1.2215881741533368,3.315182683594067), linewidth(1) + qqwwzz); draw((0.9630550791826599,0.43112970923797356)--(2.656751263931035,-1.8048003501985326), linewidth(1) + qqwwzz); draw((2.656751263931035,-1.8048003501985326)--(7.083365948870123,2.620144128539957), linewidth(1) + qqwwzz); draw((-2.4761455283243157,4.971384383012055)--(1.9504691566147727,9.396328861750545), linewidth(1) + qqwwzz); draw((-3.5720690793327403,-1.1909206061281046)--(0.8545456056063478,3.2340238726103854), linewidth(1) + qqttzz); draw((0.8545456056063478,3.2340238726103854)--(7.083365948870123,2.620144128539957), linewidth(1) + qqttzz); draw((0.8545456056063478,3.2340238726103854)--(1.9504691566147727,9.396328861750545), linewidth(1) + qqttzz); /* dots and labels */ dot((-3.5720690793327403,-1.1909206061281046),dotstyle); label("$B$", (-4.459108899614465,-0.4183375368504733), NE * labelscalefactor); dot((7.083365948870123,2.620144128539957),dotstyle); label("$C$", (7.186865515052053,2.9009082422682386), NE * labelscalefactor); dot((1.9504691566147727,9.396328861750545),dotstyle); label("$A$", (2.0649259076188664,9.68247073926078), NE * labelscalefactor); dot((3.7526748149394593,4.357504638941627),linewidth(4pt) + dotstyle); label("$H$", (2.408296160631147,4.01686156455815), NE * labelscalefactor); dot((-1.2215881741533368,3.315182683594067),linewidth(4pt) + dotstyle); label("$A_0$", (-2.3130448182877115,3.1298217442764256), NE * labelscalefactor); dot((0.9630550791826599,0.43112970923797356),linewidth(4pt) + dotstyle); label("$C_0$", (1.7787840301086324,-0.704479414360707), NE * labelscalefactor); dot((0.8545456056063478,3.2340238726103854),linewidth(4pt) + dotstyle); label("$O$", (1.1778860873371413,3.387349434035636), NE * labelscalefactor); dot((-2.4761455283243157,4.971384383012055),linewidth(4pt) + dotstyle); label("$P$", (-3.686525830336834,4.27438925431736), NE * labelscalefactor); dot((2.656751263931035,-1.8048003501985326),linewidth(4pt) + dotstyle); label("$Q$", (3.0950366666557083,-2.6216299936792735), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy][/asy]