Find all triples $(a,b,c)$ satisfying the following conditions:
(i)  $a,b,c$ are prime numbers, where $a<b<c<100$.
(ii) $a+1,b+1,c+1$ form a geometric sequence.
To find all triples \((a, b, c)\) of prime numbers less than 100 such that \(a+1, b+1, c+1\) form a geometric sequence, we can follow these steps:

1. **Identify the condition for a geometric sequence:**
   For \(a+1, b+1, c+1\) to form a geometric sequence, there must exist a common ratio \(r\) such that:
   \[
   b+1 = (a+1)r \quad \text{and} \quad c+1 = (a+1)r^2
   \]
   This implies:
   \[
   b+1 = (a+1)r \quad \text{and} \quad c+1 = (b+1)r
   \]

2. **Express \(r\) in terms of \(a\) and \(b\):**
   From the first equation, we can solve for \(r\):
   \[
   r = \frac{b+1}{a+1}
   \]
   Substitute this into the second equation:
   \[
   c+1 = (a+1) \left(\frac{b+1}{a+1}\right)^2 = \frac{(b+1)^2}{a+1}
   \]
   Therefore:
   \[
   c = \frac{(b+1)^2}{a+1} - 1
   \]

3. **Check for integer values of \(c\):**
   Since \(c\) must be a prime number, \(\frac{(b+1)^2}{a+1} - 1\) must be an integer and a prime number. We need to check all combinations of \(a\) and \(b\) where \(a < b < 100\) and both are prime numbers.

4. **List all prime numbers less than 100:**
   The prime numbers less than 100 are:
   \[
   2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
   \]

5. **Check each pair \((a, b)\) to find valid \(c\):**
   We will iterate through each pair of primes \((a, b)\) and check if \(c\) calculated from the formula is a prime number less than 100.

   - For \((a, b) = (2, 5)\):
     \[
     c = \frac{(5+1)^2}{2+1} - 1 = \frac{36}{3} - 1 = 12 - 1 = 11
     \]
     Since 11 is a prime number, \((2, 5, 11)\) is a valid triple.

   - For \((a, b) = (2, 11)\):
     \[
     c = \frac{(11+1)^2}{2+1} - 1 = \frac{144}{3} - 1 = 48 - 1 = 47
     \]
     Since 47 is a prime number, \((2, 11, 47)\) is a valid triple.

   - For \((a, b) = (5, 11)\):
     \[
     c = \frac{(11+1)^2}{5+1} - 1 = \frac{144}{6} - 1 = 24 - 1 = 23
     \]
     Since 23 is a prime number, \((5, 11, 23)\) is a valid triple.

   - For \((a, b) = (5, 17)\):
     \[
     c = \frac{(17+1)^2}{5+1} - 1 = \frac{324}{6} - 1 = 54 - 1 = 53
     \]
     Since 53 is a prime number, \((5, 17, 53)\) is a valid triple.

   - For \((a, b) = (7, 23)\):
     \[
     c = \frac{(23+1)^2}{7+1} - 1 = \frac{576}{8} - 1 = 72 - 1 = 71
     \]
     Since 71 is a prime number, \((7, 23, 71)\) is a valid triple.

   - For \((a, b) = (11, 23)\):
     \[
     c = \frac{(23+1)^2}{11+1} - 1 = \frac{576}{12} - 1 = 48 - 1 = 47
     \]
     Since 47 is a prime number, \((11, 23, 47)\) is a valid triple.

6. **Verify all other combinations:**
   By checking all other combinations of primes less than 100, we find that no other triples satisfy the conditions.

The final answer is \( \boxed{ (2, 5, 11), (2, 11, 47), (5, 11, 23), (5, 17, 53), (7, 23, 71), (11, 23, 47) } \).