which is the lowest value of c: if

$a^2 + b^2 = c^2$ , and , $ab + ac + bc = 100$

As I solve this problem with mathematica **edit**

**these conditions are generated from a right triangle with hicks a and b , and hypotenuse c is not a task , personal research**

**Replay**

```
eqns = {a^2 + b^2 == c^2, a*b + a*c + b*c == 100, a > 0, b > 0, c > 0};
sol = Minimize[{c, eqns}, {a, b, c}] // ToRadicals // FullSimplify
(* {10 Sqrt[2/7 (-1 + 2 Sqrt[2])], {a -> 10 Sqrt[1/7 (-1 + 2 Sqrt[2])],
b -> 10 Sqrt[1/7 (-1 + 2 Sqrt[2])], c -> 10 Sqrt[2/7 (-1 + 2 Sqrt[2])]}} *)
```

The approximate numerical values are

```
sol // N
(* {7.22778, {a -> 5.11081, b -> 5.11081, c -> 7.22778}} *)
```

Verifying that the solution satisfies the equations and constraints

```
eqns /. sol[[-1]]
(* {True, True, True, True, True} *)
```

**EDIT:** Since `a`

and `b`

are interchangeable this can be simplified to

```
eqns2 = {2 a^2 == c^2, a^2 + 2*a*c == 100, a > 0, c > 0};
sol2 = Minimize[{c, eqns2}, {a, c}] // ToRadicals // FullSimplify
(* {10 Sqrt[2/7 (-1 + 2 Sqrt[2])], {a -> 10 Sqrt[1/7 (-1 + 2 Sqrt[2])],
c -> 10 Sqrt[2/7 (-1 + 2 Sqrt[2])]}} *)
sol[[1]] === sol2[[1]]
(* True *)
```