# Triangle

#### MaxMontgomery

##### New member
Draw three lines through a random interior point P in triangle ABC. The three lines divide up the triangle into six smaller triangles as shown in the picture. Prove that every other triangle (shaded area) is equal to the non shaded area or you could say half the area of the ABC triangle.
I have been working on this problem for around 2 weeks now and tried many different approaches, but i ultimately couldn't prove this. I would be very thankful if someone could help me with this problem.

#### The Highlander

##### Full Member
Draw three lines through a random interior point P in triangle ABC. The three lines divide up the triangle into six smaller triangles as shown in the picture. Prove that every other triangle (shaded area) is equal to the non shaded area or you could say half the area of the ABC triangle.
I have been working on this problem for around 2 weeks now and tried many different approaches, but i ultimately couldn't prove this. I would be very thankful if someone could help me with this problem.
Assigning the letters a to f to the Areas in your triangle, as shown in the modified diagram (below), then naming the line segments around its sides as g, h, k, m, n & p and the internal line segments as u to z (again as in this diagram):-

Then g : h = a : b and v : y = f : (a+b); k : m = c : d and y : v = (c+d) : e and so on....

Does that help?

• blamocur

#### MaxMontgomery

##### New member
Im sorry i recently conducted some testing in autocad and found out that this statement is only true if its an equalatelar triangle and point P is the centroid of that triangle sorry for potentially wasting someones time.

#### MaxMontgomery

##### New member
Assigning the letters a to f to the Areas in your triangle, as shown in the modified diagram (below), then naming the line segments around its sides as g, h, k, m, n & p and the internal line segments as u to z (again as in this diagram):-

Then g : h = a : b and v : y = f : (a+b); k : m = c : d and y : v = (c+d) : e and so on....

Does that help?
Sorry, but what I was trying to prove is simply not true in general for random triangles and random points. I drew some triangles and calculated areas with the help of a program. Thanks for trying to help!

#### The Highlander

##### Full Member
Apologies for the potential confusion between the upper & lower case P's & C's; I didn't spot it until the diagram was complete and the post already in; I was perhaps too focused on avoiding letters like i and o.  #### The Highlander

##### Full Member
Sorry, but what I was trying to prove is simply not true in general for random triangles and random points. I drew some triangles and calculated areas with the help of a program. Thanks for trying to help!
No probs. If P is the centroid of an equilateral triangle that certainly makes a big difference. #### MaxMontgomery

##### New member
No probs. If P is the centroid of an equilateral triangle that certainly makes a big difference. Do you have any idea how to prove it somehow that the statement is false? Now that we know its false.

#### Dr.Peterson

##### Elite Member
Im sorry i recently conducted some testing in autocad and found out that this statement is only true if its an equalatelar triangle and point P is the centroid of that triangle sorry for potentially wasting someones time.
Do you have any idea how to prove it somehow that the statement is false? Now that we know its false.

You can prove that your initial claim (that the sums of areas are always equal) is false, by showing any counterexample. You've already done that, it seems.

But your new claim, that the sums of areas are only equal for the centroid of an equilateral triangle is false. By playing in GeoGebra, I suspect it may be true for any triangle exactly when P is on a median. I haven't bothered to prove that. Would you like to try?

• topsquark

#### MaxMontgomery

##### New member
You can prove that your initial claim (that the sums of areas are always equal) is false, by showing any counterexample. You've already done that, it seems.

But your new claim, that the sums of areas are only equal for the centroid of an equilateral triangle is false. By playing in GeoGebra, I suspect it may be true for any triangle exactly when P is on a median. I haven't bothered to prove that. Would you like to try?
Thank you for your help I will look into this
a bit more I didn't have much time yesterday so i only found this one example of the statement being true thanks for expanding it by quite a lot. So yeah I will try to prove what you hypothesized. If you have any ideas I would be thankful if you shared them here so i can come here for help if im stuck.

#### blamocur

##### Senior Member
I haven't worked it out myself, but I'd use @The Highlander's diagram in post #2 to prove that [imath]\frac{f}{e} = \frac{f+a+b}{e+d+c}[/imath]
Can you do that? Does it help to complete the solution?

#### Steven G

##### Elite Member
Draw three lines through a random interior point P in triangle ABC.
Where does the statement of your problem say that the lines must start at the vertices of the triangle???
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