Triangle Medians: Prove Area Quotient

[Hannisch]

Homework Statement

Show that a triangle's medians after suitable parallel movement (I'm directly translating from Swedish here, excuse any weird sentences) can form a new triangle. Also show that the quotation between the median triangle area and the original triangle area always are the same.

Homework Equations

The Attempt at a Solution

Okay, so I have a triangle that I've labled ABC. D is the midpoint on AB, E is the midpoint on AC and F is the midpoint on BC and then I draw the medians. I show the first problem by expressing AF, BE and CD through the other vectors and since ABC is a triangle, AB+BC+CA equals zero and I get AF+BE+CD = AB+BC+CA + ½(AB+BC+CA) = 0.

The problem comes with the second part. I know I can express the area of the original triangle in a few different ways;

Area = ½|AB×AC| = P
Area = ½|BA×BC| = P
Area = ½|CA×CB| = P

For the median triangle I can say that FA×FE is the median area, but since FE = BE, I can use that instead.

FA is the vector from the midpoint on BC to A, which can be expressed as AB + ½BC.
BE is the vector from B to the midpoint on AC, which can be expressed as BC + ½CA.

½|FA×BE| = ½|(-AB - ½BC)×(BC + ½CA)|

Since the cross product is distributive over addition and BC×BC = 0:

= ½|-AB×BC - ½BC×BC - AB×½CA - ½BC×½CA| = ½|-AB×BC -½AB×CA - (1/4)BC×CA|

So, AB = -BA. CA = -AC. BC = -CB. Thus:

= ½|-(-BA)×BC - ½AB×(-AC) - (1/4)(-CB)×CA| = ½|BA×BC + ½AB×AC + (1/4)CB×CA|

All of those three equals the area, P:

=½|2P+ P + (1/2)P| = ½|(7/2)P| = (7/4)P.

My question is: does this make sense? It seems odd that the median triangle area should be bigger than the original area.. But then again, it's not impossible - just strange.

Anyway, a yes or a no+explanation where I went wrong would be very helpful!

(should I care that the last expression is CB×CA - because when I expressed the areas in the beginning, I said CA×CB - but in this case (since it's the value we're talking about) it's equal.. even though CA×CB = - CB×CA..)

 

[mighty2000]

Yes, your solution looks correct. The reason why the median triangle's area is larger than the original triangle's area is because when we move the medians parallel to themselves, we are essentially stretching out the triangle. This increases the length of the medians and therefore increases the area of the median triangle.

As for your question about the order of the vectors in the cross product, it does not matter as long as we are consistent. So in this case, it does not matter if we use CB×CA or CA×CB. Both will give us the same result.

Overall, your solution is well thought out and presented clearly. Good job!