Question

D is a point on the side BC of \(\Delta ABC\) such that \(\angle CAB = \angle CDA\). Show that \(CA^2 = CB \times CD\).

Hint:

To avoid mistakes when writing down the ratios of corresponding sides of similar triangles, always trace the letters directly from your similarity statement:
If \(\Delta \textbf{C}\textbf{A}B \sim \Delta \textbf{C}\textbf{D}A\):
- The first two letters of both triangles give the ratio: \(\frac{CA}{CD}\)
- The first and third letters of both triangles give the ratio: \(\frac{CB}{CA}\)
Setting these equal directly yields the correct ratio without having to look back at the geometric diagram!

Answer 1

Step 1: Understanding the Question:
The topic of this question is Similar Triangles.
We are given a triangle \(ABC\) where \(D\) is a point on the side \(BC\).
We are given that the angle \(\angle CAB\) is equal to the angle \(\angle CDA\).
We need to prove the geometric relationship \(CA^2 = CB \times CD\).
To prove a multiplicative relationship among geometric segments, we should identify two triangles containing these sides and show that they are similar.

Step 2: Key Formula or Approach:
We will identify two triangles that share the vertex angle \(C\) and have another pair of equal angles:
- \(\Delta CAB\)
- \(\Delta CDA\)
We will use the AA (Angle-Angle) similarity criterion, which states that if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.
Once the similarity is established, the ratios of their corresponding sides will be equal, which will allow us to derive the required equation.

Step 3: Detailed Explanation:

• Consider the two triangles \(\Delta CAB\) and \(\Delta CDA\):
Let us list the corresponding equal angles between them:

• \(\angle CAB = \angle CDA\) (This is given to us in the problem statement).

• \(\angle ACB = \angle DCA\) (This is a common angle shared by both triangles at vertex \(C\)).

• Apply the AA (Angle-Angle) similarity criterion:
Since two corresponding pairs of angles are equal, the two triangles are similar to each other.
The correct vertex correspondence is:
\[ \Delta CAB \sim \Delta CDA \]

• Write down the ratios of their corresponding sides based on this vertex correspondence:
\[ \frac{CA}{CD} = \frac{CB}{CA} \]

• Cross-multiply the terms of the proportion:
\[ CA \cdot CA = CB \cdot CD \] \[ CA^2 = CB \times CD \] This completes our geometric proof.


Step 4: Final Answer:
Hence, it is proved that \(CA^2 = CB \times CD\).