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I'm currently wondering about the side splitter theorem.

enter image description here

Lets say I have this triangle. Now the side splitter theorem says, that if a line is parallel to a side of a triangle and intersects the two other sides, then this line divides those two sides proportionally.

In my triangle, MB is parallel to CD and intersects EC and ED.

So the theorem states that: $\frac{EB}{BD} = \frac{EM}{MC}$.

I'm wondering if the following equation is also possible: $\frac{MB}{CD} = \frac{EM}{EC}$. This equation is used in some methods of computer science (calculate corresponding screen positions), but I'm not sure, why and in which case it is correct. Does it only work because of the right angle at C ?

Thanks

Harish Chandra Rajpoot
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Frame91
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  • The side splitter theorem has nothing to do with the right angle. The conditions on the ratio shows that the triangles are similar, so you get other relations on the sides. – Calvin Lin Jul 21 '13 at 15:33
  • So my equation is possible, isn't it? Because of the similarity of both triangles (never thought about similarity, just "looked up" the equation and didn't thought about it) – Frame91 Jul 21 '13 at 15:37
  • Also known as the Intercept Theorem. A, standard, proof of this can be found here. (This does not use an argument appealing to the properties of similar triangles; however, the Intercept Theorem and the "Similar Triangles Theorem" are equivalent.) – David Mitra Jul 21 '13 at 15:48

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Triangles $ EMB $ and $ CDE $ are similiar, so yes, the last relation you mentioned holds.

Constantine
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