Hello ReneePatt Originally Posted by

**ReneePatt** 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: $\displaystyle \triangle ABC$, and $\displaystyle L, M, N$ are the mid-points of $\displaystyle BC, CA, AB$ and $\displaystyle AL \perp BC, BM \perp CA, CN\perp AB$

To prove: $\displaystyle BC = CA=AB$

Proof:

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

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

$\displaystyle AL$ is common

$\displaystyle \therefore \triangle\text{'s } ABL, ACL$ are congruent. SSS.

$\displaystyle \therefore AB = AC$

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

$\displaystyle \therefore BC = CA=AB$

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 $\displaystyle 90^o$ with the opposite side. Can you complete this now?

Grandad