Prove that in any triangle ABC, X AB vector + y AC vector = (X+Y) AD vector where D is a point on BC such that BD:DC = Y:X. Please help

To prove the vector equation \( X \vec{AB} + Y \vec{AC} = (X + Y) \vec{AD} \) in triangle ABC, where D is a point on BC such that the ratio \( BD:DC = Y:X \), we can use the concept of vector addition and the section formula. Let's break this down step by step.

Understanding the Triangle and Vectors

In triangle ABC, we can represent the position vectors of points A, B, and C as \( \vec{A} \), \( \vec{B} \), and \( \vec{C} \) respectively. The vectors \( \vec{AB} \) and \( \vec{AC} \) can be expressed as:

• \( \vec{AB} = \vec{B} - \vec{A} \)
• \( \vec{AC} = \vec{C} - \vec{A} \)

Finding the Position of Point D

Point D divides the segment BC in the ratio \( Y:X \). According to the section formula, the position vector of point D can be calculated as follows:

Let \( \vec{D} \) be the position vector of point D. Then, using the section formula, we have:

\[ \vec{D} = \frac{Y \vec{B} + X \vec{C}}{X + Y} \]

Expressing Vectors in Terms of D

Now, we can express the vectors \( \vec{AB} \) and \( \vec{AC} \) in terms of \( \vec{D} \):

Substituting \( \vec{D} \) into our original equation:

We need to show that:

\[ X \vec{AB} + Y \vec{AC} = (X + Y) \vec{AD} \]

Substituting the expressions for \( \vec{AB} \) and \( \vec{AC} \):

\[ X (\vec{B} - \vec{A}) + Y (\vec{C} - \vec{A}) = (X + Y) (\vec{D} - \vec{A}) \]

Expanding Both Sides

Now, let's expand both sides of the equation:

Left-hand side:

\[ X \vec{B} - X \vec{A} + Y \vec{C} - Y \vec{A} = X \vec{B} + Y \vec{C} - (X + Y) \vec{A} \]

Right-hand side:

\[ (X + Y) \vec{D} - (X + Y) \vec{A} \]

Substituting \( \vec{D} \) into the right-hand side gives:

\[ (X + Y) \left( \frac{Y \vec{B} + X \vec{C}}{X + Y} \right) - (X + Y) \vec{A} \]

Which simplifies to:

\[ Y \vec{B} + X \vec{C} - (X + Y) \vec{A} \]

Equating Both Sides

Now we can equate both sides:

Left-hand side:

\[ X \vec{B} + Y \vec{C} - (X + Y) \vec{A} \]

Right-hand side:

\[ Y \vec{B} + X \vec{C} - (X + Y) \vec{A} \]

Since both sides are equal, we have proven that:

\[ X \vec{AB} + Y \vec{AC} = (X + Y) \vec{AD} \]

Conclusion

This proof demonstrates how vector addition and the section formula can be applied to show the relationship between the vectors in triangle ABC. By breaking down the problem into manageable steps and using clear vector definitions, we arrive at the desired result. This approach not only solidifies our understanding of vector operations but also illustrates the beauty of geometric relationships in triangles.