Problem:
Anchored Balloon

A balloon placed on the ground is connected to one or more anchors on the ground with ropes.
Each rope is long enough to connect the balloon and the anchor.
No two ropes cross each other.
Figure E-1 shows such a situation.

Figure E-1: A balloon and ropes on the ground

Now the balloon takes off, and your task is to find how high
the balloon can go up with keeping the rope connections.
The positions of the anchors are fixed.
The lengths of the ropes and the positions of the anchors are given.
You may assume that these ropes have no weight and
thus can be straightened up when pulled to
whichever directions.
Figure E-2 shows the highest position of the balloon for
the situation shown in Figure E-1.

Figure E-2: The highest position of the balloon

Input

The input consists of multiple datasets, each in the following format.

n

x1 y1 l1

...

xn yn ln

The first line of a dataset contains an integer n (1 ≤ n ≤ 10) representing the number of the ropes.
Each of the following n lines contains three integers,
xi, yi, and li, separated by a single space.
Pi =
(xi, yi) represents
the position of the anchor connecting the i-th rope,
and li represents the length of the rope.
You can assume that
−100 ≤ xi ≤ 100,
−100 ≤ yi ≤ 100, and
1 ≤ li ≤ 300.
The balloon is initially placed at (0, 0) on the ground.
You can ignore the size of the balloon and the anchors.

You can assume that Pi and Pj
represent different positions if i ≠ j.
You can also assume that the distance between
Pi and (0, 0) is less than or equal to
li−1.
This means that the balloon can go up at least 1 unit high.

Figures E-1 and E-2 correspond to the first dataset of Sample Input below.

The end of the input is indicated by a line containing a zero.

Output

For each dataset,
output a single line containing the maximum height that the balloon can go up.
The error of the value should be no greater than 0.00001.
No extra characters should appear in the output.

Sample Input

3
10 10 20
10 -10 20
-10 10 120
1
10 10 16
2
10 10 20
10 -10 20
2
100 0 101
-90 0 91
2
0 0 53
30 40 102
3
10 10 20
10 -10 20
-10 -10 20
3
1 5 13
5 -3 13
-3 -3 13
3
98 97 168
-82 -80 193
-99 -96 211
4
90 -100 160
-80 -80 150
90 80 150
80 80 245
4
85 -90 290
-80 -80 220
-85 90 145
85 90 170
5
0 0 4
3 0 5
-3 0 5
0 3 5
0 -3 5
10
95 -93 260
-86 96 211
91 90 177
-81 -80 124
-91 91 144
97 94 165
-90 -86 194
89 85 167
-93 -80 222
92 -84 218
0

Output for the Sample Input

17.3205081
16.0000000
17.3205081
13.8011200
53.0000000
14.1421356
12.0000000
128.3928757
94.1879092
131.1240816
4.0000000
72.2251798

Solution in Rust (judge verdict: Accepted)

/*------------------------------------------------------------------*/
/*------------------------- begin template -------------------------*/
/*------------------------------------------------------------------*/

// ref: tanakh <https://qiita.com/tanakh/items/0ba42c7ca36cd29d0ac8>
// diff: using Parser
macro_rules! input {
    (source = $s:expr, $($r:tt)*) => {
        let mut parser = Parser::from_str($s);
        input_inner!{parser, $($r)*}
    };
    (parser = $parser:ident, $($r:tt)*) => {
        input_inner!{$parser, $($r)*}
    };
    (new_stdin_parser = $parser:ident, $($r:tt)*) => {
        let stdin = std::io::stdin();
        let reader = std::io::BufReader::new(stdin.lock());
        let mut $parser = Parser::new(reader);
        input_inner!{$parser, $($r)*}
    };
    ($($r:tt)*) => {
        input!{new_stdin_parser = parser, $($r)*}
    };
}

macro_rules! input_inner {
    ($parser:ident) => {};
    ($parser:ident, ) => {};
    ($parser:ident, $var:ident : $t:tt $($r:tt)*) => {
        let $var = read_value!($parser, $t);
        input_inner!{$parser $($r)*}
    };
}

macro_rules! read_value {
    ($parser:ident, ( $($t:tt),* )) => {
        ( $(read_value!($parser, $t)),* )
    };
    ($parser:ident, [ $t:tt ; $len:expr ]) => {
        (0..$len).map(|_| read_value!($parser, $t)).collect::<Vec<_>>()
    };
    ($parser:ident, chars) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()
    };
    ($parser:ident, char_) => {
        read_value!($parser, String).chars().collect::<Vec<char>>()[0]
    };
    ($parser:ident, usize1) => {
        read_value!($parser, usize) - 1
    };
    ($parser:ident, i64_) => {
        $parser.fast_i64()
    };
    ($parser:ident, usize_) => {
        $parser.fast_i64() as usize
    };
    ($parser:ident, usize1_) => {
        ($parser.fast_i64() - 1) as usize
    };
    ($parser:ident, $t:ty) => {
        $parser.next::<$t>().expect("Parse error")
    };
}

use std::io;
use std::io::BufRead;
use std::str;

// ref: tatsuya6502 <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e>
// ref: wariuni <https://qiita.com/tatsuya6502/items/cd448486f7ef7b5b8c7e#comment-7040a5ae96305e884eb9>
// diff: using std::io::BufRead::fill_buf()
pub struct Parser<R> {
    pub reader: R,
    buf: Vec<u8>,
    pos: usize,
}

impl Parser<io::Empty> {
    pub fn from_str(s: &str) -> Parser<io::Empty> {
        Parser {
            reader: io::empty(),
            buf: s.as_bytes().to_vec(),
            pos: 0,
        }
    }
}

impl<R: BufRead> Parser<R> {
    pub fn new(reader: R) -> Parser<R> {
        Parser {
            reader: reader,
            buf: vec![],
            pos: 0,
        }
    }
    pub fn update_buf(&mut self) {
        self.buf.clear();
        self.pos = 0;
        loop {
            let (len, complete) = {
                let buf2 = self.reader.fill_buf().unwrap();
                self.buf.extend_from_slice(buf2);
                let len = buf2.len();
                (len, buf2[len - 1] <= 0x20)
            };
            self.reader.consume(len);
            if complete {
                break;
            }
        }
    }
    pub fn next<T: str::FromStr>(&mut self) -> Result<T, T::Err> {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            let mut end = begin;
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return unsafe { str::from_utf8_unchecked(&self.buf[begin..end]) }.parse::<T>();
            } else {
                self.update_buf();
            }
        }
    }
    pub fn fast_i64(&mut self) -> i64 {
        loop {
            let mut begin = self.pos;
            while begin < self.buf.len() && (self.buf[begin] <= 0x20) {
                begin += 1;
            }
            if begin == self.buf.len() {
                self.update_buf();
                continue;
            }
            let mut res = 0;
            let (is_positive, mut end) = match self.buf[begin] {
                b'+' => (true, begin + 1),
                b'-' => (false, begin + 1),
                _ => (true, begin),
            };
            while end < self.buf.len() && (self.buf[end] > 0x20) {
                res = res * 10 + (self.buf[end] as i64 - '0' as i64);
                end += 1;
            }
            if begin != self.buf.len() {
                self.pos = end;
                return if is_positive { res } else { -res };
            } else {
                self.update_buf();
            }
        }
    }
}

#[allow(unused_imports)]
use std::cmp::*;

fn main() {
    input! {
        new_stdin_parser = parser,
    }
    let res = solve(&mut parser);
    print!("{}", res);
}

/*------------------------------------------------------------------*/
/*-------------------------- end template --------------------------*/
/*------------------------------------------------------------------*/

#[derive(Clone, Copy, Debug)]
pub struct Vec3 {
    x: f64,
    y: f64,
    z: f64,
}

impl Vec3 {
    pub fn new(x: f64, y: f64, z: f64) -> Vec3 {
        Vec3 { x, y, z }
    }
    pub fn dist(v: Vec3, u: Vec3) -> f64 {
        let dx = u.x - v.x;
        let dy = u.y - v.y;
        let dz = u.z - v.z;
        (dx * dx + dy * dy + dz * dz).sqrt()
    }
    pub fn to_unit(&self) -> Vec3 {
        let d = (self.x * self.x + self.y * self.y + self.z * self.z).sqrt();
        Vec3 {
            x: self.x / d,
            y: self.y / d,
            z: self.z / d,
        }
    }
}

use std::ops::*;
impl Add for Vec3 {
    type Output = Vec3;
    fn add(self, v: Vec3) -> Vec3 {
        Vec3 {
            x: self.x + v.x,
            y: self.y + v.y,
            z: self.z + v.z,
        }
    }
}

impl Mul<f64> for Vec3 {
    type Output = Vec3;
    fn mul(self, a: f64) -> Vec3 {
        Vec3 {
            x: self.x * a,
            y: self.y * a,
            z: self.z * a,
        }
    }
}

use std::fmt::Write;
fn solve<R: BufRead>(parser: &mut Parser<R>) -> String {
    let mut buf = String::new();
    loop {
        input! {
            parser = parser,
            n: usize,
        }
        if n == 0 {
            break;
        }
        input! {
            parser = parser,
            ps: [(i64,i64,i64); n],
        }
        let eps = 1e-10;
        let mut res = 0.0;
        // <1>
        'l1: for &(x1, y1, l1) in &ps {
            let v = Vec3::new(x1 as f64, y1 as f64, l1 as f64);
            for &(x2, y2, l2) in &ps {
                let u = Vec3::new(x2 as f64, y2 as f64, 0.0);
                if !(Vec3::dist(u, v) < l2 as f64 + eps) {
                    continue 'l1;
                }
            }
            if res < v.z {
                res = v.z;
            }
        }
        // <2>
        for &(x1, y1, l1) in &ps {
            'l2: for &(x2, y2, l2) in &ps {
                let d = (((x1 - x2) * (x1 - x2) + (y1 - y2) * (y1 - y2)) as f64).sqrt();
                let d1 = ((l1 * l1 - l2 * l2) as f64 + d * d) / (2.0 * d);
                let h = (l1 as f64 * l1 as f64 - d1 * d1).sqrt();
                let v = Vec3::new(x1 as f64, y1 as f64, h)
                    + Vec3::new((x2 - x1) as f64, (y2 - y1) as f64, 0.0).to_unit() * d1;
                // eprintln!("d={}, d1={}, h={}", d, d1, h);
                // eprintln!("v={:?}", v);
                for &(x4, y4, l4) in &ps {
                    let u = Vec3::new(x4 as f64, y4 as f64, 0.0);
                    if !(Vec3::dist(u, v) < l4 as f64 + eps) {
                        continue 'l2;
                    }
                }
                if res < v.z {
                    res = v.z;
                }
            }
        }
        // <3>
        for &(x1, y1, l1) in &ps {
            for &(x2, y2, l2) in &ps {
                'l1: for &(x3, y3, l3) in &ps {
                    let a = -2 * (x1 - x2);
                    let b = -2 * (y1 - y2);
                    let c = l1 * l1 - l2 * l2 - x1 * x1 + x2 * x2 - y1 * y1 + y2 * y2;
                    let d = -2 * (x1 - x3);
                    let e = -2 * (y1 - y3);
                    let f = l1 * l1 - l3 * l3 - x1 * x1 + x3 * x3 - y1 * y1 + y3 * y3;
                    let x = (c * e - b * f) as f64 / (a * e - b * d) as f64;
                    let y = (a * f - c * d) as f64 / (a * e - b * d) as f64;
                    let z = ((l1 * l1) as f64
                        - (x - x1 as f64) * (x - x1 as f64)
                        - (y - y1 as f64) * (y - y1 as f64))
                        .sqrt();
                    let v = Vec3::new(x, y, z);
                    //// eprintln!("a={}, b={}, c={}, d={}, e={}, f={}", a, b, c, d, e, f);
                    //// eprintln!("v = {:?}", v);
                    for &(x4, y4, l4) in &ps {
                        let u = Vec3::new(x4 as f64, y4 as f64, 0.0);
                        if !(Vec3::dist(u, v) < l4 as f64 + eps) {
                            continue 'l1;
                        }
                    }
                    if res < v.z {
                        res = v.z;
                    }
                }
            }
        }
        // eprintln!("res = {}", res);
        writeln!(buf, "{:.20}", res);
    }
    buf
}