If Cot A + Cot B + Cot C = √3. Then prove that triangle ABC must be equilateral

Answer 1

Probably want to pause the video and take notes.

Answer 2

Triangle ABC....A on the left, B on top, C on the right.

Drops perpendicular height, h, from B to AC, intersecting at M.

Let x = AM. Then MC = b-x.

cot A = x/h and cot C = (b-x)/h

since the cotangents are equal, x/h = (b-x)/h

so then x = b-x

2x = b

x = b/2

and b-x = b - b/2 = b/2

As a result, AM = MC = b/2, MB=BM and so

ABM and CBM are congruent right triangles.

angle A = angle C by CPCTC

The exact same argument is repeated for another side, say dropping the height from A

to BC. The result shall be the same, and angle B = angle C

and then again, to get angle A = angle B

Since angles A=B=C, they must be 60 degrees each, which proves the equilateral.