Finding the minimum area of a triangle

[Null_Pointer]

So i have the points A=(1,0,1) and B=(1,-1,0) and the third corner lies on the line
L:(x,y,z) = (t,t,t) and i need to find a triangle with the minimum area possible.

My initial approach was to calculate the vector BA and BC and then apply both vectors in the area triangle formula 1/2||BAxBC|| = area and then find a t that gives me the minimum area, but then I am wondering if this is correct since there may be a possibility that t can be canceled out and I'm left with a constant that comes from the derivate of the cross product.

 

[Some Pig]

There is a simpler way, since A,B are fixed points, the area of the triangle is minimum if the third point is closest to AB. This is equivalent to find the distance between L and AB.

 

[HallsofIvy]

So i have the points A=(1,0,1) and B=(1,-1,0) and the third corner lies on the line
L:(x,y,z) = (t,t,t) and i need to find a triangle with the minimum area possible.

My initial approach was to calculate the vector BA and BC and then apply both vectors in the area triangle formula 1/2||BAxBC|| = area and then find a t that gives me the minimum area, but then I am wondering if this is correct since there may be a possibility that t can be canceled out and I'm left with a constant that comes from the derivate of the cross product.

Why would you worry about that possiblity? It doesn't cancel!