Geometric Proof: Proving Independence of DE + DF in Isosceles Triangle ABC

In summary: Sorry, I left out a factor. The sum of the two lengths is equal to ##\frac {2ha}{\sqrt{a^2+h^2}}## where ##h## is the height of the triangle and ##2a## is the length of the base. So the units are area/length = length. But, as I said, not the proof the OP is looking for.
  • #1
DotKite
81
1

Homework Statement


In triangle ABC, AB = AC, and D,E,F are points on the interiors of sides BC,AB,AC respectively, such that DE perpendicular to AB and DF perpendicular to AC. Prove that the value of DE + DF is independent of the location of D


Homework Equations


So far we have all the tools of neutral geometry and non neutral parallelism. We have not covered similarity yet


The Attempt at a Solution



Ok so I guess a good approach would be to consider triangle ABC with D in one location and then another and show there is no change in DE + DF. However I do not have a clue in how to proceed with this. Tried using the fact that triangle ABC is isosceles therefore the base angles are equal, but don't really know where to go with that either
 
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  • #2
DotKite said:
In triangle ABC, AB = AC, and D,E,F are points on the interiors of sides BC,AB,AC respectively, such that DE perpendicular to AB and DF perpendicular to AC. Prove that the value of DE + DF is independent of the location of D

I don't think that's true.

Suppose D is either B or the midpoint of BC.

Then the perpendicular from B to AC would have to be twice the perpendicular from that midpoint.

But it isn't, is it? :confused:
 
  • #3
tiny-tim said:
I don't think that's true.

Suppose D is either B or the midpoint of BC.

Then the perpendicular from B to AC would have to be twice the perpendicular from that midpoint.

But it isn't, is it? :confused:

Yes, I think it is. Draw the midline parallel to AC. It will bisect BF.

[Edit, added]: It's true alright, and easy enough to prove analytically. The sum of DE and DF comes out equal to the length of the base divided by the length of one of the equal legs, independent of the location of D. Not that any of this helps the OP. :frown:
 
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  • #4
@OP, can you post a picture it will truly help.
 
  • #5
LCKurtz said:
Yes, I think it is. Draw the midline parallel to AC. It will bisect BF.

[Edit, added]: It's true alright, and easy enough to prove analytically. The sum of DE and DF comes out equal to the length of the base divided by the length of one of the equal legs, independent of the location of D. Not that any of this helps the OP. :frown:

not following you :confused:

(how can the sum of two lengths equal the ratio of two lengths, ie a number?)

anyway, don't forget that you can't use most of the usual theorems about parallel lines
 
  • #6
tiny-tim said:
not following you :confused:

(how can the sum of two lengths equal the ratio of two lengths, ie a number?)

anyway, don't forget that you can't use most of the usual theorems about parallel lines

Sorry, I left out a factor. The sum of the two lengths is equal to ##\frac {2ha}{\sqrt{a^2+h^2}}## where ##h## is the height of the triangle and ##2a## is the length of the base. So the units are area/length = length. But, as I said, not the proof the OP is looking for.
 

1. What is a geometric proof?

A geometric proof is a method of showing that a statement or theorem is true using logical arguments based on previously proven statements and accepted rules of geometry.

2. What does it mean to prove independence of DE + DF in an isosceles triangle?

In this context, proving independence of DE + DF means showing that the lengths of segments DE and DF are not dependent on each other, and can be treated as separate, individual lengths within the triangle.

3. Why is it important to prove independence of DE + DF in an isosceles triangle?

Proving independence of DE + DF is important because it allows us to make accurate conclusions and calculations about the triangle without any assumptions or restrictions on the relationship between the two segments. It also allows us to generalize our findings to other isosceles triangles.

4. What are some common methods used in geometric proof?

Some common methods used in geometric proof include using definitions, postulates, and theorems, as well as logical reasoning and constructing diagrams or drawings to visually represent the given information.

5. How can I improve my skills in writing geometric proofs?

To improve your skills in writing geometric proofs, it is important to have a strong understanding of geometric concepts and properties. Practice writing proofs using different methods and techniques, and seek feedback from others. Additionally, studying and analyzing examples of well-written proofs can also help improve your skills.

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