Circle

(1,−2)=(α,−α−1)
⇒α=1
It is clear from question that one of the vertex of triangle is intersection of x-axis and x+y+1=0⇒A(−1,0)
Let vertex B be (α,−α−1)
Line AC⟂BH so, mAC⋅mBH=−1
⇒O=−‌

(1−α)

α+2

⇒α=1⇒B(1,−2)
Let vertex C be (β,0)
Line AH⟂BC
∴mAH−mBC=−1
⇒‌

1

2

⋅‌

2

β−1

=−1⇒‌‌β=0
Centroid of △ABC is (0,−‌

2

3

)
We know that G (centroid) divides line joining circumcentre (O) and orthocentre (H) in the ratio 1:2.⇒ Circumcentre is (−‌

1

2

,−‌

3

2

).
Equation of circum circle is (passing through C(0,0) ) is x2+y2+x+3y=0