A great puzzle!
We have a circle with an equilateral triangle inscribed in it . We join the mid points (A and B) of the triangle and let the midline extend to meet the circle at a point C on the same side as B. Then find the ratio AB:BC.Give your answer correct to 5 decimal places.
Try to solve with 10th grade maths.
The answer is 1.61803.
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1 solution
A and B are midpoints of any two sides. Let a be the length of one side.
Join A, B, C, and O ( centre of circle). In ∆AOC, we know angle OAC = 30°, we know OC= radius of circle = √ 3 a , and we know AO= inradius= 2 √ 3 a . We can now use the cosine rule in triangles to find out the third side, AC= AB + BC= ½a + BC. (AB = 0.5a through midpoint theorem).
BC comes out to be 4 a ( √ 5 − 1 )
Thus, AB:BC = √ 5 − 1 2 ≈ 1 . 6 1 8 .