<s>[INST] (BAMO $2013/3$) $ABH$, $BCH$ and $CAH$ is congruent to $ABC$. 
Let $H$ be the orthocenter of an acute triangle $ABC$. Prove that the triangle formed by the circumcenters of triangles $ABH$, $BCH$ and $CAH$ is congruent to $ABC$.

I have already seen many answers on MSe But my doubt is diffferent,
In this solution (first one )
https://artofproblemsolving.com/community/c618937h1628954_problem_320_bamo_20133
To prove $O_AB || O_BA$, we can do some angle calculations: $\angle O_ABC = 90 - A$, and $\angle C A O_B = 90 - B$
how he got $\angle O_ABC = 90 - A$, and $\angle C A O_B = 90 - B$ ?
I tried some angle chasing but did not able to get this ..thankyou
 [/INST] Because $AO_B=CO_B$, $\measuredangle AO_BC=2\beta$ and from here:
$$\measuredangle CAO_B=90^{\circ}-\beta.$$
Another way.
Let $A'$, $B'$ and $C'$ be circumcenters of $\Delta BHC$, $\Delta CHA$ and $\Delta AHB$ respectively.
Thus, since $B'HA'C$ is a rhombus and $HC=c\cot\gamma$ in the standard notation, we obtain:
$$A'B'=2\sqrt{B'H^2-\left(\frac{HC}{2}\right)^2}=2\sqrt{R^2-\frac{1}{4}c^2\cot^2\gamma}=$$
$$=2\sqrt{\frac{a^2b^2c^2}{16S^2}-\frac{1}{4}c^2\cdot\frac{\left(\frac{a^2+b^2-c^2}{2ab}\right)^2}{\frac{4S^2}{a^2b^2}}}=2\sqrt{\frac{a^2b^2c^2}{16S^2}-\frac{c^2(a^2+b^2-c^2)^2}{64S^2}}=$$
$$=\frac{c}{4S}\cdot\sqrt{4a^2b^2-(a^2+b^2-c^2)^2}=\frac{c}{4S}\cdot\sqrt{\sum_{cyc}(2a^2b^2-a^4)}=c.$$
Similarly $A'C'=b$ and $B'C'=a$  and we are done!
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