Question

Let I, O, H be the incenter, circumcenter, and orthocenter of an acute triangle ABC, respectively. Prove that if the points B, C, H, I lie on a single circle, then O lies on this circle too.

1. The points B, C, H, I lie on a single circle, then O lies on this circle too.
For given question,
We have been given I, O, H be the incenter, circumcenter, and orthocenter of an acute triangle ABC, respectively
In an acute ΔABC, O is circumcenter. Thus, by applying the inscribed angle theorem to the circle, we have:
∠BOA = 2∠BAC.               ………………(1)
Assuming H is orthocenter; Thus, the angle formed by angles BAC and BC at H are supplementary angles:
∠BAC + BHC = 180°.
BHC = 180° – ∠BAC.           ……………….(2)
Assuming I is incenter; Thus, the angle formed by BIC is given by:
∠BIC = 90° + ∠A/2.            ……………….(3)
Since points B, C, H, I lie on a single circle and the angles formed in the same side of an arc of a circle are equal, we have:
BIC = BHC                           ……………….(4)
Substituting the parameters into equation (4), we have;
⇒ 180° – ∠BAC = 90° + ∠A/2
⇒ ∠BAC + ∠BAC/2 – 3∠BAC/2 = 180° – 90°
⇒ ∠BAC = 90° × 2/3
⇒ ∠BAC = 60°.
From equation (1), we have:
⇒ ∠BOA = 2∠BAC.
⇒ ∠BOA = 2 × 60
⇒ ∠BOA = 120°.
Hence, this proves that since points B, C, H, I lie on a single circle, then O lies on this circle too.