How do you prove this triangle to be equilateral?

A circle centre O has 3 points on its circumference, A, B and C, such that ABC is an equilateral triangle. Point D lies on the circumference of the circle such that OD bisects AB. Prove triangle ODA is equilateral.

1 Answer
May 9, 2018

see explanation.

Explanation:

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Given that #DeltaABC# is an equilateral triangle, #=> OA# bisects #angleBAC, => angleOAB=60/2=30^@#
given that #OD# bisects chord #AB, => M# is the midpoint of chord #AB#,
recall that the line joining the center of a circle to the midpoint of a chord is perpendicular to the chord,
#=> OM# is perpendicular to #AB#,
#=> angleOMA=90^@, => angleMOA=180-90-30=60^@#
as #OA=OD=r, => angleODA=angleOAD=(180-angleDOA)/2=(180-60)/2=60^@#,
As #angleODA=angleOAD=angleDOA=60^@#,
#=> DeltaODA# is equilateral.