e.cpp

#include<iostream>
#include<complex>
#include<cstdio>

using namespace std;

#define rep(i, n) for (int i = 0; i < int(n); ++i)

typedef long double D;
typedef complex<D> P;
struct L{P a, b;};

#define X real()
#define Y imag()

D det(P a, P b) {return a.X * b.Y - a.Y * b.X;}
P vec(L a) {return a.b - a.a;}

P pLL(L a, L b) {return a.a + vec(a) * (det(vec(b), b.a - a.a) / det(vec(b), vec(a)));}

int n;
P pp[11];
D l[11];
const D EPS = 1e-8;

D mysr(D a) {return sqrt(max(a, (D)0));}

bool can(P p, D h) {
  rep (i, n) {
    if (mysr(norm(p - pp[i]) + pow(h, 2)) > l[i] + EPS) {
      return false;
    }
  }
  return true;
}

int main() {
  while (true) {
    cin >> n;
    if (n == 0) break;
    rep (i, n) {
      cin >> pp[i].X >> pp[i].Y >> l[i];
    }
    D res = 0;
    rep (i, n) rep (j, i) {
      D theta = acos((pow(l[i], 2) + norm(pp[i] - pp[j]) - pow(l[j], 2)) / (2 * l[i] * abs(pp[i] - pp[j])));
      P btm = pp[i] + (pp[j] - pp[i]) / abs(pp[j] - pp[i]) * l[i] * cos(theta);
      D h = l[i] * sin(theta);
      if (can(btm, h)) res = max(res, h);
    }
    rep (i, n) {
      if (can(pp[i], l[i])) res = max(res, l[i]);
    }
    rep (i, n) rep (j, i) rep (k, j) {
      D theta1 = acos((pow(l[i], 2) + norm(pp[i] - pp[j]) - pow(l[j], 2)) / (2 * l[i] * abs(pp[i] - pp[j])));
      P btm1 = pp[i] + (pp[j] - pp[i]) / abs(pp[j] - pp[i]) * l[i] * cos(theta1);
      D theta2 = acos((pow(l[i], 2) + norm(pp[i] - pp[k]) - pow(l[k], 2)) / (2 * l[i] * abs(pp[i] - pp[k])));
      P btm2 = pp[i] + (pp[k] - pp[i]) / abs(pp[k] - pp[i]) * l[i] * cos(theta2);
      L l1 = (L){btm1, btm1 + (pp[i] - pp[j]) * P(0, 1)};
      L l2 = (L){btm2, btm2 + (pp[i] - pp[k]) * P(0, 1)};
      P ppp = pLL(l1, l2);
      if (!can(ppp, 0)) continue;
      D h = mysr(l[i] * l[i] - norm(pp[i] - ppp));
      if (can(ppp, h)) res = max(res, h);
    }
    printf("%.12Lf\n", res);
  }
  return 0;
}