In , , , and . A circle cuts at and , cuts at and , and cuts at and . If , and that the area of the hexagon formed by the vertices and is , find .
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Our circle has the triangle incentre as its centre. Radius of the incircle is 1. Points of contact of incircle with triangle sides are midpoints of segments A 1 A 2 , B 1 B 2 , C 1 C 2 . This makes segment lengths A C 1 = A B 2 = ( 3 − 1 ) − 2 x = 2 − 2 x , B A 1 = B C 2 = 1 − 2 x , C A 2 = C B 1 = 3 − 2 x .
△ A C 1 B 2 = 2 1 ( 2 − 2 x ) 2 sin ( A ) = 2 1 ( 2 − 2 x ) 2 5 4 △ B A 1 C 2 = 2 1 ( 1 − 2 x ) 2 sin ( B ) = 2 1 ( 1 − 2 x ) 2 △ C A 2 B 1 = 2 1 ( 3 − 2 x ) 2 sin ( C ) = 2 1 ( 3 − 2 x ) 2 5 4
△ A C 1 B 2 + △ B A 1 C 2 + △ C A 2 B 1 = △ A B C − 4 = 2
This gives value of x = 3 1 1 − 3 7 ≈ 1 . 6 4