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Geometry Level pending

If the segment A T AT is the bisector of angle α \alpha (angle at point A A ), is the following statement true or false ?

B T × sin ⁡ γ = C T × sin ⁡ β BT \times \sin{\gamma} = CT \times \sin{\beta}

False True

Milan Milanic by Milan Milanic , 24, Serbia

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1 solution

Milan Milanic
Jan 20, 2016

Solution:

A T sin ⁡ β = B T sin ⁡ α 2 \frac{AT}{\sin{\beta}} = \frac{BT}{\sin{\frac{\alpha}{2}}} and A T sin ⁡ γ = C T sin ⁡ α 2 \frac{AT}{\sin{\gamma}} = \frac{CT}{\sin{\frac{\alpha}{2}}} ( sine theorem )

Therefore, A T = sin ⁡ β sin ⁡ α 2 B T = sin ⁡ γ sin ⁡ α 2 C T AT = \frac{\sin{\beta}}{\sin{\frac{\alpha}{2}}} BT = \frac{\sin{\gamma}}{\sin{\frac{\alpha}{2}}} CT . Which is C T × sin ⁡ γ = B T × sin ⁡ β CT \times \sin{\gamma} = BT \times \sin{\beta} .

The given statement is slightly different, but those differences are important and therefore, the statement is invalid .

P.S. If someone has an idea for the problem's name, write a comment. As the title (at this current time) suggests, I don't have so much ideas.

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