While doing geometry problem I encountered something I would call "inverse" of inscribed angle theorem.

Garrett Sheppard

Garrett Sheppard

Open question

2022-08-15

"Inverse" of the inscribed angle theorem.
While doing geometry problem I encountered something I would call "inverse" of inscribed angle theorem. At first I wanted to state it like this:
Given ABC and point O such that BOC = 2 BAC it implies that O is the centre of the circle described on the ABC
But of course it is easy to show that this statement doesn't hold. But on my native language forum I found that this theorem should hold:
Given ABC and A = α and point O lying on the perpendicular bisector of the segment BC with BOC = 2 α and point O lying on the same side of line BC as point A then point O is the centre of the circle described on ABC

Answer & Explanation

Erika Brady

Erika Brady

Beginner2022-08-16Added 19 answers

Step 1
Suppose there is a point O O satisfying the conditions.

Step 2
Then C B O = C B O = 90 α so O B O = 0 and then O = O .

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