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.

Answers

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.

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