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Math Help - Proving equilaterial triangle using centroid and circumcenter

  1. #1
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    Proving equilaterial triangle using centroid and circumcenter

    I am asked to: Prove that a triangle is equilaterial if and only if its centroid and circumcenter coincide.

    I know that:

    (a) centroid of a triangle is the point where the three medians (a segment joining a vertex of a triangle to the midpoint of the opposite side) of the triangle intersect.

    (b) circumcenter of a triangle is the point where the three pependicular bisectors (a line segment that is both perpendicular to a side of a triangle and passes through its midpoint) of a triangle intersect.

    But I don't know how to do the proof or what definitions/theorems are used for the proof.
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  2. #2
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    Hello ReneePatt
    Quote Originally Posted by ReneePatt View Post
    I am asked to: Prove that a triangle is equilaterial if and only if its centroid and circumcenter coincide.

    I know that:

    (a) centroid of a triangle is the point where the three medians (a segment joining a vertex of a triangle to the midpoint of the opposite side) of the triangle intersect.

    (b) circumcenter of a triangle is the point where the three pependicular bisectors (a line segment that is both perpendicular to a side of a triangle and passes through its midpoint) of a triangle intersect.

    But I don't know how to do the proof or what definitions/theorems are used for the proof.
    You have correctly identified the characteristics of the centroid and the circumcentre. You need to prove this in both directions.

    First if the centroid and circumcentre coincide, prove that the triangle is equilateral. So:
    Given: \triangle ABC, and L, M, N are the mid-points of BC, CA, AB and AL \perp BC, BM \perp CA, CN\perp AB

    To prove: BC = CA=AB

    Proof:

    In \triangle\text{'s } ABL, ACL:
    BL = LC (given)

     \angle ALB = \angle ALC (=90^o, given )

     AL is common
    \therefore \triangle\text{'s } ABL, ACL are congruent. SSS.

     \therefore AB = AC

    Similarly in \triangle\text{'s } BCM, BAM, BC = BA

     \therefore BC = CA=AB
    Second we prove that if a triangle is equilateral, then the centroid and circumcentre coincide. Do this using congruent triangles to prove that each median makes an angle of 90^o with the opposite side. Can you complete this now?

    Grandad
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  3. #3
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    Quote Originally Posted by Grandad View Post
    First if the centroid and circumcentre coincide, prove that the triangle is equilateral. So:
    Given: \triangle ABC, and L, M, N are the mid-points of BC, CA, AB and AL \perp BC, BM \perp CA, CN\perp AB

    To prove: BC = CA=AB

    Proof:

    In \triangle\text{'s } ABL, ACL:
    BL = LC (given)

     \angle ALB = \angle ALC (=90^o, given )

     AL is common
    \therefore \triangle\text{'s } ABL, ACL are congruent. SSS.

     \therefore AB = AC

    Similarly in \triangle\text{'s } BCM, BAM, BC = BA

     \therefore BC = CA=AB
    Second we prove that if a triangle is equilateral, then the centroid and circumcentre coincide. Do this using congruent triangles to prove that each median makes an angle of 90^o with the opposite side. Can you complete this now?

    Grandad

    Hi Grandad,

    How do you have the givens? Are you assuming the givens because centroid and circumcenter are supposed to coincide?

    Quote Originally Posted by Grandad View Post
    BL = LC (given)

     \angle ALB = \angle ALC (=90^o, given )
    Also, how are you getting that
    BL = LC (given)

     \angle ALB = \angle ALC (=90^o, given )

    are given?
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  4. #4
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    Thumbs up I think I've got it

    Grandad, I think I've got it - Thanks
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