A triangle is drawn with its vertices on the circle $$C$$ such that one of its sides is a diameter of $$C$$ and the other two sides have their lengths in the ratio $$a : b$$. If the radius of the circle is $$r$$, then the area of the triangle is

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1. **Triangle Properties:**  
   Since one side of the triangle is the diameter of the circle, the triangle is a right-angled triangle (angle subtended by the diameter on the circle is 90°). Let the lengths of the other two sides be $$ax$$ and $$bx$$, where $$x$$ is a scaling factor.

2. **Area Formula:**  
   The area of the triangle is:  
   $$
   	ext{Area} = \frac{1}{2} \cdot (	ext{Base}) \cdot (	ext{Height}) = \frac{1}{2} \cdot ax \cdot bx = \frac{abx^2}{2}.
   $$

3. **Using Pythagoras Theorem:**  
   For a right-angled triangle:  
   $$
   (ax)^2 + (bx)^2 = (Diameter)^2 = (2r)^2.
   $$  
   Simplify:  
   $$
   a^2x^2 + b^2x^2 = 4r^2.
   $$  
   Factorize:  
   $$
   x^2(a^2 + b^2) = 4r^2.
   $$  
   Solve for $$x^2$$:  
   $$
   x^2 = \frac{4r^2}{a^2 + b^2}.
   $$

4. **Substitute $$x^2$$ into Area Formula:**  
   $$
   	ext{Area} = \frac{abx^2}{2} = \frac{ab}{2} \cdot \frac{4r^2}{a^2 + b^2}.
   $$  
   Simplify:  
   $$
   	ext{Area} = \frac{2abr^2}{a^2 + b^2}.
   $$

Thus, the area of the triangle is **$$\frac{2abr^2}{a^2 + b^2}$$**.

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