Results 1 to 15 of 15

Math Help - [SOLVED] construction of equilateral triangle

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    55

    [SOLVED] construction of equilateral triangle

    Can someone help me with this:
    construct equilateral triangle if vertex A is given and vertices B and C belong to given lines b i c.

    Thanks.
    Attached Thumbnails Attached Thumbnails [SOLVED] construction of equilateral triangle-example.jpg  
    Last edited by Garas; January 18th 2010 at 05:47 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    If vertex A is given, then you have it's co-ordinates.

    For B, label the point (x_2,y_2)

    For C, label the point (x_3,y_3)

    now use the line equations to write one variable in terms of the other
    in both cases.

    Then use Pythagoras' theorem since the distance between A and B
    equals the distance from A to C equals the distance from B to C.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Jan 2010
    Posts
    55
    I can't use coordinate system so i don't have coordinates of vertex A.I think that isometric transformations(rotation) have to be used to solve this but i'm not sure.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Yes,
    sorry i thought the problem involved given the line equations and point A.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jan 2010
    Posts
    1
    Does anyone know how to solve this?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    The circles are for construction of isosceles triangles.
    Their common centre is the given point.
    An equilateral triangle is a special case of isosceles triangle.
    Hence the use of the purple line with a 120 degree angle as shown.
    We only need a parallel line to this from the point to find one triangle side.
    the blue parallel lines find the other two.
    Attached Thumbnails Attached Thumbnails [SOLVED] construction of equilateral triangle-equilateral-triangle.jpg  
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Junior Member
    Joined
    Jan 2010
    Posts
    55
    Why do you need two blue lines and circles?
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    The blue lines are parallel,
    the red line cuts all the blue segments between the green given lines at their midpoints.
    i use the circles since every point on the circle is equidistant from the given point,
    therefore the diagram is creating isosceles triangles with the given point as vertex.
    Only one circle is needed really, i use 2 for emphasis.
    The problem is to find the isosceles triangle that is also equilateral.
    Hence, the purple line is used to locate the 120 degree angle,
    meaning the inner angle is 60 degrees.
    Then the parallel dotted lines discover an isosceles triangle with two 60 degree angles,
    hence the 3rd angle is also 60 degrees, therefore it's equilateral.
    That is my thinking on it.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Junior Member
    Joined
    Jan 2010
    Posts
    55
    Can you prove that?Check this image, it shows that your construction isn't OK.
    Attached Thumbnails Attached Thumbnails [SOLVED] construction of equilateral triangle-graphic2.jpg  
    Last edited by Garas; January 19th 2010 at 07:16 AM.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Your lower dotted purple line should be adjusted to intersect the green
    and blue lines, giving three 60 degree angles.

    I understand your concern,
    i have to go for a few hours,
    but i will double-check everything later.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    My diagram only works if the point is on the bisector of the angle between the 2 given lines.
    I'll try to find the solution to the general case.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Let A' be the reflection of A in the line c. Draw a line through A' making an angle 30 with AA', meeting the line b at Y. Let X be the midpoint of AY. Draw a line through X, perpendicular to AY, meeting the line c at Z. Then the triangle AYZ is equilateral.

    Reason: the construction ensures that the triangles AA'Z and AYZ are isosceles, so that A'Z = AZ = YZ. Thus the points A', A, Y all lie on a circle with centre Z. Thus angle AZY is 60 (angle at centre = twice angle at circumference). It follows that the isosceles triangle AZY is equilateral.

    There is a second equilateral triangle with a vertex at A and the other vertices on b and c, obtained by drawing the line A'Y at an angle of 30 on the other side of AA' (to the left of AA' instead of to the right as in my picture).
    Attached Thumbnails Attached Thumbnails [SOLVED] construction of equilateral triangle-equilat.png  
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Junior Member
    Joined
    Jan 2010
    Posts
    55
    That is excellent.Thanks
    Last edited by Garas; January 19th 2010 at 02:15 PM.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    If the angle between the lines is bisected and the perpendicular bisector drawn, followed by drawing the perpendicular to this from the given point
    then we can send 2 lines at 30 degrees to the bisector, from the point of intersection of the 2 construction lines.
    The green triangle is isosceles at the other end, so it is equilateral also.

    Therefore, using the given point as the circle centre, we can draw the black circle, then use the given point as centre, draw a circle of the exact same radius.
    This is the dashed circle.
    Where this circle intersects the lines gives the other 2 triangle vertices.
    Attached Thumbnails Attached Thumbnails [SOLVED] construction of equilateral triangle-equilateral-triangle2.jpg  
    Follow Math Help Forum on Facebook and Google+

  15. #15
    MHF Contributor
    Joined
    Dec 2009
    Posts
    3,120
    Thanks
    1
    Here's another view, using rotating equilateral triangles
    and parallelograms.
    Attached Files Attached Files
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. equilateral triangle help
    Posted in the Geometry Forum
    Replies: 3
    Last Post: September 1st 2011, 06:29 PM
  2. Equilateral Triangle
    Posted in the Geometry Forum
    Replies: 3
    Last Post: October 21st 2010, 04:37 AM
  3. Equilateral triangle
    Posted in the Geometry Forum
    Replies: 2
    Last Post: March 10th 2010, 11:23 AM
  4. Replies: 1
    Last Post: May 18th 2009, 10:31 PM
  5. Equilateral Triangle
    Posted in the Geometry Forum
    Replies: 8
    Last Post: October 5th 2008, 10:58 AM

Search Tags


/mathhelpforum @mathhelpforum