Show There Is No Triangle With Altitudes 1 2 3

Proving the Impossibility of a Triangle with Altitudes 1, 2, and 3

Meta Description: This article provides a mathematical proof demonstrating that no triangle can exist with altitudes of length 1, 2, and 3. We explore the area relationships and inequalities involved to reach a conclusive result.

It's a fascinating geometrical question: can a triangle exist with altitudes of lengths 1, 2, and 3? Intuitively, it might seem possible, but a rigorous mathematical proof reveals otherwise. This article will demonstrate that such a triangle is impossible.

Understanding Altitudes and Area

Before we delve into the proof, let's refresh our understanding of a triangle's altitude. An altitude is a perpendicular line segment from a vertex to the opposite side (or its extension). Crucially, the area of a triangle can be calculated using any side as the base and the corresponding altitude:

Area = (1/2) * base * height

This simple formula will be key to our proof.

The Proof by Contradiction

We'll use proof by contradiction. We'll assume a triangle with altitudes a=1, b=2, and c=3 exists, and then show this assumption leads to a contradiction.

Let's denote the sides of the triangle as x, y, and z, corresponding to altitudes a, b, and c respectively. The area of the triangle can be expressed in three ways:

• Area = (1/2) * x * 1
• Area = (1/2) * y * 2
• Area = (1/2) * z * 3

This gives us three equations:

• Area = x/2
• Area = y
• Area = 3z/2

Since all three expressions represent the same area, we can equate them:

• x/2 = y = 3z/2

From these equations, we can derive relationships between the sides:

• x = 2y
• z = y/3

Now, let's consider the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This gives us three inequalities:

1. x + y > z
2. x + z > y
3. y + z > x

Substituting our expressions for x and z in terms of y:

1. 2y + y > y/3 => 3y > y/3 (This inequality always holds true)
2. 2y + y/3 > y => 7y/3 > y (This inequality always holds true)
3. y + y/3 > 2y => 4y/3 > 2y => 4 > 6 (This is a contradiction!)

The third inequality leads to a false statement (4 > 6). This contradiction arises directly from our initial assumption that a triangle with altitudes 1, 2, and 3 exists. Therefore, our assumption is false, and no such triangle can exist.

Conclusion

Through a straightforward application of the area formula and the triangle inequality theorem, we've conclusively shown that it's impossible to construct a triangle with altitudes of length 1, 2, and 3. This highlights the importance of understanding fundamental geometric principles and the power of proof by contradiction in mathematical reasoning. The seemingly simple question reveals a deeper mathematical truth.