Question

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.

Answer 1

see explanation.

Explanation:

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.