Example 1:

If , prove that a, b, c, and d are proportionals.

Solution 1:

Given that

We also know that if , then the property of componendo and dividendo says: . Applying this property to the above “huge” fraction, we get:

We know that if , then , which is the property called alternendo. Applying this property to the above fraction, we get

,

again, applying componendo and dividendo to the above, we get

hence, , that is, a, b, c and d are proportionals. Hence, the proof.

Example 2:

Solve the equation:

Solution 2:

By componendo and dividendo, we get

Now, squaring both sides of the above equation, we get .

Again, applying componendo and dividendo,

so, we get

so, we get finally

Cheers,

Nalin Pithwa