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Equilateral triangle ABC is inscribed in a circle

  1. Nov 29, 2014 #1
    1. The problem statement, all variables and given/known data
    Can someone help me solve this, and teach me how to solve such problems in future? An equilateral triangle ABC is inscribed in a circle . Point D lies on a shorter arc of a circle BC. Point E is symmetrical the point B relating to the line CD . Prove that the points A, D , E lie on one straight line. If something is unclear just tell me.

    2. Relevant equations
    --------

    3. The attempt at a solution

    I do not know how to go about Himself .
    Frame I figure .
     

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  2. jcsd
  3. Nov 29, 2014 #2

    ehild

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    Welcome to PF!


    Use the Theorem of Inscribed Angles.
     
  4. Nov 29, 2014 #3

    HallsofIvy

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    Excellent English! Much better than before. Congratulations. This may be more than is necessary but I am more comfortable with "analysis" than "geometry" so I would set up a coordinate system with origin at the center of the circle and (0, 1) at point C. Then the equation of the circle, in that coordinate system, is [itex]x^2+ y^2= 1[/itex]. On can show that the line through the point A and the center of the circle is [itex]y= x/2[/itex] so that point A has coordinates where that line intersects the circle. With [itex]y= x/2[/itex] the equation of the circle becomes [itex]x^2+ x^2/4= (5/4)x^2= 1[/itex] so that [itex]x= -\frac{2}{\sqrt{5}}= -\frac{2\sqrt{5}}{5}[/itex] and [itex]y= -\frac{\sqrt{5}}{5}[/itex]. Similarly for point B. With that information it should be fairly easy to find the coordinates of points D and E and show that they line on a single line.
     
  5. Nov 29, 2014 #4
    I figured all the angles. What next?
     

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  6. Nov 29, 2014 #5

    ehild

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    Try to determine the angle ADE. (First find the angle of the blue right triangle at D.

    circleangles.JPG
     
  7. Nov 29, 2014 #6
    How do I determine the angle ADE as it is simple? .
    Calculated the angles of the blue triangle.
    Help ! "D l
     
  8. Nov 29, 2014 #7

    ehild

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    You have to prove that ADE is a straight line, that is, the angle ADE is 180°. What are the angles ABC, ADC ADB,?
    The red line CD halves the blue triangle, as it is symmetric to the red line. Wjhat are the angles at D?
     
  9. Nov 29, 2014 #8
    All I calculated.
    I can't write request.
     

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  10. Nov 29, 2014 #9

    ehild

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    Show what you calculated. What are the angles γ and δ; β and λ? How many degrees? What is the angle σ?

    circleangles.JPG
     
  11. Nov 29, 2014 #10
    Already pasted.
    circleangles-jpg.75935.jpg 29-listopad-2014-2-gif.75931.gif
     

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  12. Nov 29, 2014 #11

    RUber

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    You seem to be assuming that the line drawn from A will intersect BC at a 90 degree angle. This is incorrect.
    In post #2, ehild said to use the law of inscribed angles. Check that out and see if you can find all the angles in terms of the one variable angle.
     
  13. Nov 30, 2014 #12

    ehild

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    @dawo0: your drawing does not help to solve the problem. It shows the special case when the point D is at the middle of the arc BC. But you need to prove the statement of the problem for any point D on the arc.
    You have drawn ADE as a straight line, but you have to prove that it is a straight line!
    Do you understand why are γ=δ=60°and λ=β=60°? You should explain. In general, you should explain your statements when solving a problem.
    Something like that: β is one angle of the equilateral triangle, so it is 60°. β and λ are inscribed angles of the circle, belonging to the same arc AC, so λ=β...

    29-listopad-2014-2-gif.75931.gif [/QUOTE]
     
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