ThatDonGuy
ThatDonGuy
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January 15th, 2019 at 7:34:24 PM permalink
All-American Gaming's logo consists of two equilateral triangles and a circle, as shown:



If AB has length 1, then what is the length of AO?
ChesterDog
ChesterDog
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January 15th, 2019 at 10:34:02 PM permalink
Quote: ThatDonGuy

All-American Gaming's logo consists of two equilateral triangles and a circle, as shown:



If AB has length 1, then what is the length of AO?



half the square root of 21--I'm surprised how hard this problem is.
ThatDonGuy
ThatDonGuy
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January 16th, 2019 at 6:44:44 PM permalink
Quote: ChesterDog

half the square root of 21--I'm surprised how hard this problem is.


That's what I get as well. What surprised me was, the distance from A to where circle O is tangent to line AC is a rational number.
ChesterDog
ChesterDog
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January 16th, 2019 at 9:30:17 PM permalink
Quote: ThatDonGuy

Quote: ChesterDog

half the square root of 21--I'm surprised how hard this problem is.


That's what I get as well. What surprised me was, the distance from A to where circle O is tangent to line AC is a rational number.



That is interesting!

I looked for an easy way to see it. Think of a 60-degree rhombus with vertices (0, 0), (1, 0), (1/2, 31/2/2), and (3/2, 31/2/2). Then inscribe a circle in the rhombus. The center of the circle is at the midpoint of the line segment connecting (0, 0) and (3/2, 31/2/2), which is (3/4, 31/2/4).

So, that rational number distance between A and where circle O is tangent to line AC is 2 + 3/4 = 11/4.

(edit) I have the rhombus slanting toward the right, but in your diagram it slants to the left. So, the distance in your diagram would be 2+1/4 = 9/4.
Last edited by: ChesterDog on Jan 16, 2019
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