Language: C++

#include<bits/stdc++.h>
using namespace std;

#define FOR(i,bg,ed) for(int i=(bg);i<(ed);i++)
#define REP(i,n) FOR(i,0,n)
#define ALL(v) (v).begin(),(v).end()
#define fi first
#define se second
#define pb push_back

#define IL inline 
namespace geo {
    typedef double D;
    typedef bool B;
    typedef complex<D> P;
#define X real()
#define Y imag()
    B comp(const P& l, const P& r){ return (l.X == r.X) ? l.Y < r.Y : l.X < r.X; }
    typedef pair<P, P> L;  //line
    typedef pair<P, P> LS; //line segment
    typedef pair<P, D> C;  //circle
    typedef vector<P> Poly;
    const D EPS = 1e-8;
    //Decompotision Macro
#define DCl(a,b,l) P (a),(b);tie(a,b)=l; 
#define DCc(a,b,c) P (a);D (b);tie(a,b)=c; 
    
    //A dot B
    IL D dot(P a, P b) { return a.X*b.X + a.Y*b.Y; }
    //A cross B
    IL D cross(P a, P b) { return a.X*b.Y - a.Y*b.X; }
    IL D ccw(P a,P b,P c){
        return cross(b - a, c - a);
    }

    IL int sgn(P a,P b, P c){
        if(cross(b-a,c-a) > EPS)return 1;
        if(cross(b-a,c-a) < -EPS)return -1;
        if(dot(b-a,c-a) < -EPS)return 2; // b-a-c
        if(dot(a-b,c-b) < -EPS)return -2; // a-b-c
        return 0;
    }
    IL D signed_distance_P_L(P p, L l) {
        DCl(a,b,l);
        return ccw(a,b,p)/ abs(b - a);
    }
    
    IL D distance_P_L(P p, L l) {
        return abs(signed_distance_P_L(p,l));
    }

    IL D distance_P_LS(P p, LS ls) {
        DCl(a,b,ls);
        if (dot(b - a, p - a) < EPS) return abs(p - a);
        if (dot(a - b, p - b) < EPS) return abs(p - b);
        return abs(cross(b - a, p - a)) / abs(b - a);
    }

    
    IL B is_L_L(L p, L q) {
        DCl(a,b,p);
        DCl(c,d,q);
        return
            abs(cross(a - b, c - d)) > EPS ||
            abs(cross(a - b ,d - b)) < EPS;
    }
    IL B is_L_LS(L l, LS ls) {
        DCl(a,b,l);
        DCl(c,d,ls);
        return ccw(a,b,c)*ccw(a,b,d)<EPS;
    }
    IL B is_LS_LS(LS p, LS q) {
        DCl(a,b,p);
        DCl(c,d,q);
        return sgn(a,b,c)*sgn(a,b,d) <= 0
            && sgn(c,d,a)*sgn(c,d,b) <= 0;
    }

    IL D distance_LS_LS(LS p,LS q){
        if(is_LS_LS(p,q))return 0;
        return min({
                distance_P_LS(p.fi,q),
                distance_P_LS(p.se,q),
                distance_P_LS(q.fi,p),
                distance_P_LS(q.se,p)    
                    });
    }

    //caution: have to use isintersected_L_L before this function
    IL P cross_point_L_L(L p, L q) {
        DCl(a,b,p);
        D d1 = signed_distance_P_L(a, q);
        D d2 = signed_distance_P_L(b, q);
        return (a * d2 - b * d1) / (d2 - d1);
    }
    
    IL B is_C_C(C p, C q) {
        DCc(cp,rp,p);
        DCc(cq,rq,q);
        D d = abs(cp - cq);
        return abs(rp - rq) - d <EPS &&
                                 rp + rq - d > -EPS;
    }
    IL LS cross_C_C(C p, C q) {
        DCc(cp,rp,p);
        DCc(cq,rq,q);
        D d = abs(cp - cq);
        D rc = (d*d + rp*rp - rq*rq) / (2 * d);
        D rs = sqrt(rp*rp - rc*rc);
        P diff = (cq - cp) / d;
        return {cp + P(rc, rs)*diff, cp + P(rc, -rs)*diff};
    }
    IL D ccw(P a,P b,P c){
        return cross(b - a, c - a);
    }
    IL Poly convex_hull(Poly& p) {
        int n = p.size();
        Poly res(2 * n);
        int k = 0, t;
        sort(ALL(p),comp);
        REP(i,n) {
            while (k > 1 && ccw(res[k-2],res[k-1],p[i]) < EPS)k--;
            res[k++] = p[i];
        }
        t = k;
        REP(j,n-1){
            int i= n - 2 - j;
            while (k > t && ccw(res[k - 2],res[k - 1], p[i]) < EPS)k--;
            res[k++] = p[i];
        }
        res.resize(k - 1);
        return res;
    }
    IL B convex_in(Poly& l,P& p) {
        int a = 0, b = l.size(), c;
        D A, C;
        P g = (l[a] + l[b - 1] + l[b / 2]) / 3.0;
        while (b - a > 1) {
            c = (a + b) / 2;
            A = cross(l[a] - g, p - l[a]);
            C = cross(l[c] - g, p - l[c]);
            if (cross(l[a] - g, l[c] - g) >= 0) {
                if (A > -EPS&&C < -EPS)b = c;
                else a = c;
            }
            else {
                if (C < -EPS || A > -EPS)b = c;
                else a = c;
            }
        }
        return(cross(l[b%l.size()] - l[a], p - l[b%l.size()]) > -EPS);
    }
    IL D convex_area(Poly& l) {
        double res = 0;
        int n = l.size();
        FOR(i,2,n)
            res += (abs(cross(l[i - 1] - l[0], l[i] - l[i - 1]))) / 2.0;
        return res;
    }
    inline vector<L> tangent_line(C c, P p) {
        vector<L> res;
        DCc(cp,r,c);
        P v = p - cp;
        D dist = abs(v);
        if (dist - r > -EPS) {
            double _cos = r / dist;
            double _sin = sqrt(1.0 - _cos*_cos);
            v = v * (r / dist);
            {
                P e = P(_cos * v.X - _sin * v.Y, _sin*v.X + _cos*v.Y);
                res.push_back({p, cp + e});
            }
            if(dist-r > EPS){
                P e = P( _cos*v.X + _sin*v.Y ,-_sin*v.X + _cos*v.Y );
                res.push_back({p, cp + e});
            }
        }
        return res;
    }
    IL vector<L> inner_common_tangent_line(C p, C q) {
        vector<L> res;
        DCc(Mp,Rp,p);
        DCc(Mq,Rq,q);
        D dist = abs(Mp - Mq);
        if (dist - (Rp + Rq) > -EPS) {
            P mid = (Mq*Rp + Mp*Rq) / (Rp + Rq);
            res = tangent_line(p, mid);
        }
        return res;
    }
    inline vector<L> outer_common_tangent_line(C p, C q) {
        vector<L> res;
        DCc(Mp,Rp,p);
        DCc(Mq,Rq,q);
        P v = Mp - Mq;
        D dist = abs(v);
        if (dist - abs(Rp - Rq) >- EPS) {
            if (abs(Rp - Rq) < EPS) {
                P e = v*P( 0, 1 );
                e = e / dist;
                res.pb(L(Mp + e, Mq + e));
                res.pb(L(Mp - e, Mq - e));
            }
            else {
                P mid = Mp + (Mq - Mp) * Rp / (Rp - Rq);
                res = tangent_line(p, mid);
            }
        }
        return res;
    }
}
int main(){
    int a1, b1, r1;
    int a2, b2, r2;
    cin >> a1 >> b1 >> r1;
    cin >> a2 >> b2 >> r2;
    auto v = geo::inner_common_tangent_line(C(P(a1, b1), r1), C(P(a2, b2), r2));
    auto v2= geo::outer_common_tangent_line(C(P(a1, b1), r1), C(P(a2, b2), r2));
    cout << v.size() + v2.size() << endl;

    return 0;
}

Compiler output:
a.cc:113:10: error: redefinition of 'geo::D geo::ccw(P, P, P)'
  113 |     IL D ccw(P a,P b,P c){
      |          ^~~
a.cc:32:10: note: 'geo::D geo::ccw(P, P, P)' previously defined here
   32 |     IL D ccw(P a,P b,P c){
      |          ^~~
a.cc: In function 'int main()':
a.cc:217:47: error: 'P' was not declared in this scope; did you mean 'geo::P'?
  217 |     auto v = geo::inner_common_tangent_line(C(P(a1, b1), r1), C(P(a2, b2), r2));
      |                                               ^
      |                                               geo::P
a.cc:15:24: note: 'geo::P' declared here
   15 |     typedef complex<D> P;
      |                        ^
a.cc:217:45: error: 'C' was not declared in this scope; did you mean 'geo::C'?
  217 |     auto v = geo::inner_common_tangent_line(C(P(a1, b1), r1), C(P(a2, b2), r2));
      |                                             ^
      |                                             geo::C
a.cc:21:24: note: 'geo::C' declared here
   21 |     typedef pair<P, D> C;  //circle
      |                        ^