Vrijeme: 18:42

Dali smo mu ime! | We gave it a name! #5

Neka je ABC raznostranični trokut. Njemu upisana kružnica dodiruje stranice BC, AC i AB u točkama A_1, B_1 i C_1 redom. Pripisana kružnica nasuprot vrha A dodiruje pravce BC, AC i AB u točkama A_2, B_2 i C_2, respektivno. Pretpostavimo da se pravci AA_1, BB_1, and CC_1 sijeku u točki G i da se pravci AA_2, BB_2, and CC_2 sijeku u točki G'.Pravac GG' siječe unutarnju simetralu kuta \angle BAC u točki T. Pretpostavimo da je AT=1, \cos{\angle BAC}=\sqrt{3}-1 i da je BC=8\sqrt[4]{3}. Nađi AB \cdot AC. Odgovor zaokruži na 4 decimale.
Let ABC be a scalene triangle. The circle inscribed in it touches the sides BC, AC and AB at the points A_1, B_1 and C_1 respectively. The escribed circle opposite to vertex A touches lines BC, AC, and AB at points A_2, B_2, and C_2, respectively. Assume that lines AA_1, BB_1, and CC_1 intersect at point G and that lines AA_2, BB_2, and CC_2 intersect at point G'. Line GG' intersects the interior bisector of angle \angle BAC at point T. Assume that AT=1, \cos{\angle BAC}=\sqrt{3}-1 and that BC=8\sqrt[4]{3}. Find AB \cdot AC. Round the answer to 4 decimal places.