Problem:
Projection For given three points p1, p2, p , find the projection point x of p onto p1p2 .   Input  x p1  y p1  x p2  y p2  q  x p 0  y p 0  x p 1  y p 1 ... x p q−1  y p q−1  In the first line, integer coordinates of p1 and p2 are given. Then, q queries are given for integer coordinates of p . Output For each query, print the coordinate of the projection point x . The output values should be in a decimal fraction with an error less than 0.00000001. Constraints   1 ≤ q ≤ 1000    -10000 ≤ x i , y i ≤ 10000    p1 and p2 are not identical.  Sample Input 1 0 0 2 0
3
-1 1
0 1
1 1 Sample Output 1 -1.0000000000 0.0000000000
0.0000000000 0.0000000000
1.0000000000 0.0000000000 Sample Input 2 0 0 3 4
1
2 5 Sample Output 2 3.1200000000 4.1600000000

Solution (Python):
import sys
input = sys.stdin.readline


class Vector():
    def __init__(self, x, y):
        self.x = x
        self.y = y

    def __add__(self, vec):
        return Vector(self.x+vec.x, self.y+vec.y)

    def __sub__(self, vec):
        return Vector(self.x-vec.x, self.y-vec.y)

    def __mul__(self, sc):
        return Vector(self.x*sc, self.y*sc)

    def __truediv__(self, sc):
        return Vector(self.x/sc, self.y/sc)

    def __iadd__(self, vec):
        self.x += vec.x
        self.y += vec.y
        return self

    def __isub__(self, vec):
        self.x -= vec.x
        self.y -= vec.y
        return self

    def __imul__(self, sc):
        self.x *= sc
        self.y *= sc
        return self

    def __itruediv__(self, sc):
        self.x /= sc
        self.y /= sc
        return self

    def __str__(self):
        return f'{self.x} {self.y}'

    def __eq__(self, vec):
        return self.x == vec.x and self.y == vec.y

    def dot(self, vec):
        return self.x * vec.x + self.y * vec.y

    def abs(self):
        return (self.x*self.x + self.y*self.y)**0.5


x1, y1, x2, y2 = map(int, input().split())
v1 = Vector(x1, y1)
v2 = Vector(x2, y2)
n = v2-v1
n /= n.abs()

for _ in [0]*int(input()):
    x, y = map(int, input().split())
    v = Vector(x, y) - v1
    print(v1 + n * (n.dot(v)))

Solution (Python):
pt = []
x1, y1, x2, y2 = map(int, input().split())
q = int(input())

for _ in range(q):
    xq, yq = map(int, input().split())
    if x2 - x1 == 0:
        pt.append([x1, yq])
    else:
        a1 = (y2-y1)/(x2-x1)
        b1 = y1 - a1*x1
        if a1 == 0:
            pt.append([xq, y1])
        else:
            a2 = -1/a1
            b2 = yq - a2*xq
            x = (b2-b1)/(a1-a2)
            y = a1*x + b1
            pt.append([x, y])

for i in range(q):
    print(pt[i][0], pt[i][1])

Solution (Python):
EPS = 10**(-9)
def is_equal(a,b):
    return abs(a-b) < EPS

def norm(v,i):
    import math
    ret = 0
    n = len(v)
    for j in range(n):
        ret += abs(v[j])**i
    return math.pow(ret,1/i)

class Vector(list):
    """
    ベクトルクラス
    対応演算子
    +  : ベクトル和 
    -  : ベクトル差
    *  : スカラー倍、または内積
    ** : 外積
    += : ベクトル和
    -= : ベクトル差
    *= : スカラー倍
    /= : スカラー倍

    メソッド
    self.norm(i) : L{i}ノルムを計算
    """
    def __add__(self,other):
        n = len(self)
        ret = [0]*n
        for i in range(n):
            ret[i] = super().__getitem__(i) + other.__getitem__(i)
        return self.__class__(ret)
    
    def __radd__(self,other):
        n = len(self)
        ret = [0]*n
        for i in range(n):
            ret[i] = other.__getitem__(i) + super().__getitem__(i)
        return self.__class__(ret)
    
    def __iadd__(self, other):
        n = len(self)
        for i in range(n):
            self[i] += other.__getitem__(i)
        return self

    def __sub__(self,others):
        n = len(self) 
        ret = [0]*n
        for i in range(n):
            ret[i] = super().__getitem__(i) - others.__getitem__(i)
        return self.__class__(ret)

    def __iadd__(self, other):
        n = len(self)
        for i in range(n):
            self[i] -= other.__getitem__(i)
        return self

    def __rsub__(self,others):
        n = len(self) 
        ret = [0]*n
        for i in range(n):
            ret[i] = others.__getitem__(i) - super().__getitem__(i)
        return self.__class__(ret)
    
    def __mul__(self,other):
        n = len(self)
        if isinstance(other,list):
            ret = 0
            for i in range(n):
                ret += super().__getitem__(i)*other.__getitem__(i)
            return ret
        else:
            ret = [0]*n
            for i in range(n):
                ret[i] = super().__getitem__(i)*other
            return self.__class__(ret)

    def __rmul__(self,other):
        n = len(self)
        if isinstance(other,list):
            ret = 0
            for i in range(n):
                ret += super().__getitem__(i)*other.__getitem__(i)
            return ret
        else:
            ret = [0]*n
            for i in range(n):
                ret[i] = super().__getitem__(i)*other
            return self.__class__(ret)
    
    
    def __truediv__(self,other):
        """
        ベクトルのスカラー除法
        Vector/scalar
        """
        n = len(self)
        ret = [0]*n
        for i in range(n):
            ret[i] = super().__getitem__(i)/other
        return self.__class__(ret)
    
    def norm(self,i):
        """
        L{i}ノルム
        self.norm(i)
        """
        return norm(self,i)
    
    def __pow__(self,other):
        """
        外積
        self**other
        """
        n = len(self)
        ret = [0]*3
        x = self[:]
        y = other[:]
        if n == 2:
            x.append(0)
            y.append(0)
        if n == 2 or n == 3:
            for i in range(3):
                ret[0],ret[1],ret[2] = x[1]*y[2]-x[2]*y[1],x[2]*y[0]-x[0]*y[2],x[0]*y[1]-x[1]*y[0]
            ret = Vector(ret)
            if n == 2:
                return ret.norm(2)
            else:
                return ret

class Segment:
    def __init__(self,v1,v2):
        self.v1 = v1
        self.v2 = v2

class Line(Segment):
    def get_unit_vec(self):
        return (self.v2-self.v1)/norm(self.v2-self.v1,2)
    
    def projection(self,vector):
        unit_vec = self.get_unit_vec()
        t = unit_vec*(vector-self.v1)
        return self.v1 + t*unit_vec
    
    def is_vertical(self,other):
        return is_equal(0,self.get_unit_vec()*other.get_unit_vec())
    
    def is_horizontal(self,other):
        return is_equal(0,self.get_unit_vec()**other.get_unit_vec())



tmp = list(map(int,input().split()))
p1 = Vector(tmp[:2])
p2 = Vector(tmp[2:])
L = Line(p1,p2)
q = int(input())
for i in range(q):
    p = Vector(map(int,input().split()))
    print(*L.projection(p))

Solution (Python):
from decimal import Decimal as D
a,s,d,f=map(D,input().split())
h=[d-a,f-s]
leh=((d-a)**2+(f-s)**2)**(D("0.5"))
for i in range(int(input())):
    x,y=map(D,input().split())
    co=(x-a)*h[0]/leh+(y-s)*h[1]/leh
    print(a+h[0]*co/leh,s+h[1]*co/leh)

Solution (Python):
class Vec2:
    def __init__(self, y, x):
        self.y = y
        self.x = x

    def __add__(self, other):
        assert isinstance(other, Vec2)
        return Vec2(self.y+other.y, self.x+other.x)

    def __sub__(self, other):
        assert isinstance(other, Vec2)
        return Vec2(self.y - other.y, self.x - other.x)

    def __mul__(self, other):
        if isinstance(other, Vec2):
            return self.y * other.y + self.x * other.x
        else:
            return Vec2(self.y * other, self.x * other)

    def __str__(self):
        return str((self.y, self.x))

    def size(self):
        return (self.x**2 + self.y**2)**0.5

    def projection(self, vec):
        p = self*vec/self.size()**2
        return self*p


def main():
    x_p1, y_p1, x_p2, y_p2 = map(int, input().split())
    p1 = Vec2(y_p1, x_p1)
    p2 = Vec2(y_p2, x_p2)
    p1p2 = p2-p1
    q = int(input())
    for _ in range(q):
        x_q, y_q = map(int, input().split())
        p1q = Vec2(y_q, x_q) - p1
        p1x = p1p2.projection(p1q)
        x = p1 + p1x
        print(x.x, x.y)


if __name__ == "__main__":
    main()

Solution (Python):
class Point:
    def __init__(self, x, y):
        self.x = x
        self.y = y
    def __add__(self, other):
        return Point(self.x + other.x, self.y + other.y)
    def __sub__(self, other):
        return Point(self.x - other.x, self.y - other.y)
    def __mul__(self, other):
        return Point(self.x * other, self.y * other)
    def __floordiv__(self, other):
        return Point(self.x / other, self.y / other)
    def __repr__(self):
        return str(self.x) + ' ' + str(self.y)

class Vector(Point):
    pass

class Segment:
    def __init__(self, p1, p2):
        self.p1 = p1
        self.p2 = p2

def points_to_vector(p1, p2):
    x = p1.x - p2.x
    y = p1.y - p2.y
    return Vector(x, y)

def vector(p):
    return Vector(p.x, p.y)

def dot(v1, v2):
    return v1.x * v2.x + v1.y * v2.y

def cross(v1, v2):
    return v1.x * v2.y - v1.y * v2.x

def norm(v):
    return v.x**2 + v.y**2

def project(s, p):
    base = points_to_vector(s.p1, s.p2)
    hypo = points_to_vector(p, s.p1)
    r = dot(hypo, base) / norm(base)
    return s.p1 + base * r


import sys


s = list(map(int, input().split()))
p1 = Point(s[0], s[1])
p2 = Point(s[2], s[3])
segment = Segment(p1, p2)

q = int(input())
for i in range(q):
    x, y = map(int, input().split())
    p = Point(x, y)
    ans = project(segment, p)
    print(ans)

Solution (Python):
import math

class Point:
    def __init__(self,x,y):
        self.x = x
        self.y = y

    def __add__(self,a):
        return self.__class__(self.x+a.x,self.y+a.y)

    def __sub__(self,a):
        return self.__class__(self.x-a.x,self.y-a.y)

    def __mul__(self,d):
        return self.__class__(self.x*d,self.y*d)

    def __truediv__(self,d):
        return self.__class__(self.x/d,self.y/d)

    def __str__(self):
        return '%.10f %.10f'%(self.x,self.y)

    def abs(self):
        return self.norm()**(1.0/2.0)

    def norm(self):
        return (self.x*self.x+self.y*self.y)

    def rotate(self,d):
        r = math.pi*d/180.0
        return self.__class__( self.x*math.cos(r)-self.y*math.sin(r),
                               self.x*math.sin(r)+self.y*math.cos(r))

    def dot(self,p):
        return (self.x*p.x+self.y*p.y)

class Segment:
    def __init__(self,p1,p2):
        self.p1 = p1
        self.p2 = p2

def reflect(s,p):
    v = s.p2-s.p1
    r = v.dot(p-s.p1)/v.norm()
    return s.p1+v*r

def main():
    a,b,c,d = map(float,input().split())
    s = Segment(Point(a,b),Point(c,d))
    q = int(input())
    for _ in range(q):
        e,f = map(float,input().split())
        p = Point(e,f)
        print (reflect(s,p))


if __name__ == '__main__':
    main()

Solution (Python):
a,b,c,d=map(float,input().split())
c-=a;d-=b
for _ in [0]*int(input()):
    x,y=map(float,input().split())
    x-=a;y-=b
    z=(c*x+d*y)/(c*c+d*d)
    print(a+c*z,b+d*z)

Solution (Python):
import math

EPS = 1e-10

class Segment():
    def __init__(self, p1, p2):
        self.p1 = p1
        self.p2 = p2

class Point():
    def __init__(self, x, y):
        self.x = x
        self.y = y

    def __add__(self, point):
        
        return Point(self.x+point.x, self.y+point.y)

    def __sub__(self, point):
        
        return Point(self.x-point.x, self.y-point.y)

    def __mul__(self, a):
        
        return Point(a*self.x, a*self.y)

    def __truediv__(self, a):
        
        return Point(self.x/a, self.y/a)

    def __abs__(self):
        
        return math.sqrt(self.norm())

    def norm(self):
        return self.x**2+self.y**2

    def __eq__(self, point):
        
        return abs(self.x-point.x) < EPS and abs(self.y-point.y) <EPS

def dot(a, b):
    return a.x*b.x+a.y*b.y

def cross(a, b):
    return a.x*b.y - a.y*b.x

def isOrthogonal(a, b):
    return dot(a, b) == 0

def isParallel(a, b):
    return cross(a, b) == 0

def project(s, p):
    
    base = s.p2 - s.p1
    r = dot(p-s.p1, base)/base.norm()
    return s.p1+base*r

if __name__ == '__main__':
    from sys import stdin
    input = stdin.readline

    x1, y1, x2, y2 = map(int, input().split())
    s = Segment(Point(x1, y1), Point(x2, y2))

    q = int(input())
    for _ in range(q):
        x0, y0 = map(int, input().split())
        p = Point(x0, y0)
        projection = project(s, p)
        print(projection.x, projection.y)

Solution (Python):
from math import pi, cos, sin, atan2
EPS = 10**(-9)

def eq(value1, value2):
    return abs(value1-value2) <= EPS

class Point():
    def __init__(self, x, y):
        self.x = x
        self.y = y
        self.arg = atan2(y, x) 
    
    def __str__(self):
        return "{0:.8f} {1:.8f}".format(self.x, self.y)
    
    def __add__(self, other):
        return Point(self.x + other.x, self.y + other.y)
    
    def __sub__(self, other):
        return Point(self.x - other.x, self.y - other.y)
    
    def __mul__(self, scal):
        return Point(self.x*scal, self.y*scal)
    
    def __truediv__(self, scal):
        return Point(self.x/scal, self.y/scal)
    
    def __eq__(self, other):
        return eq(self.x, other.x) and eq(self.y, other.y)

    
    def __abs__(self):
        return (self.x**2+self.y**2)**0.5
    

def Rotation(vec: Point, rad):
    return Point(vec.x*cos(rad)-vec.y*sin(rad), vec.x*sin(rad)+vec.y*cos(rad))


class Circle():
    def __init__(self, p, r):
        self.p = p
        self.r = r


class Line():
    
    def __init__(self, a, b):
        self.a = a
        self.b = b
        self.arg = (a-b).arg % pi
    
    def __str__(self):
        return "[({0}, {1}) - ({2}, {3})]".format(self.a.x, self.a.y, self.b.x, self.b.y)

    
    def par(self, point):
        return Line(point, point+(self.a-self.b))

    
    def tan(self, point):
        return Line(point, point + Rotation(self.a-self.b, pi/2))


class Segment(Line):
    def __init__(self, a, b):
        super(Line, self).__init__(a, b)



def cross(vec1: Point, vec2: Point):
    return vec1.x*vec2.y - vec1.y*vec2.x


def dot(vec1: Point, vec2: Point):
    return vec1.x*vec2.x + vec1.y*vec2.y


def ccw(a, b, c):
    if cross(b-a, c-a) > EPS: return +1 
    if cross(b-a, c-a) < -EPS: return -1 
    if dot(c-a, b-a) < -EPS: return +2 
    if abs(b-a) < abs(c-a): return -2 
    return 0 



def projection(l, p):
    t = dot(l.b-l.a, p-l.a) / abs(l.a-l.b)**2
    return l.a + (l.b-l.a)*t


def reflection(l, p):
    return p + (projection(l, p) - p)*2

def isPararell(l1, l2):
    return eq(cross(l1.a-l1.b, l2.a-l2.b), 0)

def isVertical(l1, l2):
    return eq(dot(l1.a-l1.b, l2.a-l2.b), 0)


def isIntersect_lp(l, p):
    return abs(ccw(l.a, l.b, p)) != 1

def isIntersect_ll(l1, l2):
    return not isPararell(l1, l2) or isIntersect_lp(l1, l2.a)

def isIntersect_sp(s, p):
    return ccw(s.a, s.b, p) == 0

def isIntersect_ss(s1, s2):
    return ccw(s1.a, s1.b, s2.a)*ccw(s1.a, s1.b, s2.b) <= 0 and ccw(s2.a, s2.b, s1.a)*ccw(s2.a, s2.b, s1.b) <= 0

def isIntersect_ls(l, s):
    return cross(l.b - l.a, s.a - l.a) * cross(l.b - l.a, s.b - l.a) < EPS

def isIntersect_cp(c, p):
    return abs(abs(c.p - p) - c.r) < EPS

def isIntersect_cl(c, l):
    return distance_lp(l, c.p) <= c.r + EPS

def isIntersect_cs(c, s):
    pass

def isIntersect_cc(c1, c2):
    pass


def distance_pp(p1, p2):
    return abs(p1-p2)

def distance_lp(l, p):
    return abs(projection(l,p)-p)

def distance_ll(l1, l2):
    return 0 if isIntersect_ll(l1, l2) else distance_lp(l1, l2.a)

def distance_sp(s, p):
    r = projection(s, p)
    if isIntersect_sp(s, r): return abs(r-p)
    return min(abs(s.a-p), abs(s.b-p))

def distance_ss(s1, s2):
    if isIntersect_ss(s1, s2): return 0
    return min([distance_sp(s1, s2.a), distance_sp(s1, s2.b), distance_sp(s2, s1.a), distance_sp(s2, s1.b)])

def distance_ls(l, s):
    if isIntersect_ls(l, s): return 0
    return min(distance_lp(l, s.a), distance_lp(l, s.b))


def crosspoint_ll(l1, l2):
    A = cross(l1.b - l1.a, l2.b - l2.a)
    B = cross(l1.b - l1.a, l1.b - l2.a)
    if eq(abs(A), 0) and eq(abs(B), 0): return l2.a
    return l2.a + (l2.b - l2.a) * B / A

def crosspoint_ss(s1, s2):
    return crosspoint_ll(s1, s2)

def crosspoint_lc(l, c):
    if eq(distance_lp(l, c.p), c.r): return [c.p]
    p = projection(l, c.p)
    e = (l.b - l.a) / abs(l.b-l.a)
    dis = (c.r**2-abs(p-c.p)**2)**0.5
    return [p + e*dis, p - e*dis]

def crosspoint_sc(s, c):
    pass

def crosspoint_cc(c1, c2):
    d = abs(c1.p-c2.p)
    if not abs(c1.r-c2.r) <= d <= c1.r+c2.r:
        return []
    mid_p = (c2.p * (c1.r**2-c2.r**2+d**2) + c1.p * (c2.r**2-c1.r**2+d**2)) / (2*d**2)
    tanvec = Rotation(c1.p-c2.p, pi/2)
    return crosspoint_lc(Line(mid_p, mid_p+tanvec), c1)



def tangent_cp(c, p):
    return crosspoint_cc(c, Circle(p, (abs(p-c.p)**2 - c.r**2)**0.5))





import sys
input = sys.stdin.readline

def verify_1A():
    p1x, p1y, p2x, p2y = map(int, input().split())
    l = Line(Point(p1x, p1y), Point(p2x, p2y))
    Q = int(input())
    Query = [list(map(int, input().split())) for _ in range(Q)]
    for px, py in Query:
        p = Point(px, py)
        print(projection(l, p))

verify_1A()

Solution (Python):
def project(x,y):
    base=[x2-x1,y2-y1]
    hypo=[x-x1,y-y1]
    r=(base[0]*hypo[0]+base[1]*hypo[1])/(base[0]**2+base[1]**2)
    return x1+base[0]*r,y1+base[1]*r


x1,y1,x2,y2=map(int,input().split())
q=int(input())
for i in range(q):
    x,y=map(int,input().split())
    print(*project(x,y))

Solution (Python):
def dot(a,b):
    return a.real*b.real+a.imag*b.imag

x1, y1, x2, y2 = map(int, input().split())
p1=x1+y1*1j
p2=x2+y2*1j
l=p2-p1
l_norm=dot(l,l)         
q=int(input())

for x3, y3 in (map(int, input().split()) for _ in [0]*q):
    p3=x3+y3*1j
    a=p3-p1
    k=dot(a,l)/l_norm
    projection_vector=k*l
    print(x1+projection_vector.real,y1+projection_vector.imag)

Solution (Python):
def projection(x1, y1, x2, y2, x0, y0):
    if x2 == x1:
        return (x2, y0)
    m = float(y2 - y1) / float(x2 - x1)
    b = y1 - m * x1 
    xp = (x0 + (m * y0) - (m * b)) / (1. + (m * m))
    yp = ((m * x0) + ((m * m) * y0) + b) / (1. + (m * m))
    return (xp, yp)

x1, y1, x2, y2 = map(float, input().split())
q = int(input())
for _ in range(q):
    x0, y0 = map(float, input().split())
    ans = projection(x1, y1, x2, y2, x0, y0)
    print(ans[0], ans[1])

Solution (Python):
EPS = 0.0000000001

class Point:
    
    global EPS
    
    def __init__(self, x = 0, y = 0):
        self.x = x
        self.y = y
        
        
    def __add__(a, b):
        s = a.x + b.x
        t = a.y + b.y
        return Point(s, t)
        
    def __sub__(a, b):
        s = a.x - b.x
        t = a.y - b.y
        return Point(s, t)
            
    def __mul__(self, a):
        s = a * self.x
        t = a * self.y
        return Point(s, t)
        
    def __truediv__(self, a):
        s = self.x / a
        t = self.y / a
        return Point(s, t)
            
            
            
            
    def norm(self):
        return self.x * self.x + self.y * self.y
        
    def abs(self):
        return self.norm() ** 0.5
            
    
            
            
    def __eq__(self, other):
        return abs(self.x - other.y) < self.EPS and abs(self.y - other.y) < self.EPS
            
            
            
    def dot(self, b):
        return self.x * b.x + self.y * b.y
        
    def cross(self, b):
        return self.x * b.y - self.y * b.x
    
    
class Segment:
    def __init__(self, p1, p2):
        self.p1 = p1
        self.p2 = p2

def project(s, p):
    base = s.p2 - s.p1
    hypo = p - s.p1
    r = hypo.dot(base) / base.norm()
    return s.p1 + base * r



nums=list(map(int,input().split()))
s = Segment(Point(nums[0], nums[1]), Point(nums[2], nums[3]))

n = int(input())
for i in range(n):
    nums=list(map(int,input().split()))
    p = Point(nums[0], nums[1])
    x = project(s, p)
    print(x.x, x.y)

Solution (Python):
import cmath
import os
import sys

if os.getenv("LOCAL"):
    sys.stdin = open("_in.txt", "r")

sys.setrecursionlimit(10 ** 9)
INF = float("inf")
IINF = 10 ** 18
MOD = 10 ** 9 + 7


PI = cmath.pi
TAU = cmath.pi * 2


class Point:
    EPS = 1e-9

    def __init__(self, c: complex):
        self.c = c

    @property
    def x(self):
        return self.c.real

    @property
    def y(self):
        return self.c.imag

    @staticmethod
    def from_rect(x: float, y: float):
        return Point(complex(x, y))

    @staticmethod
    def from_polar(r: float, phi: float):
        return Point(cmath.rect(r, phi))

    def __add__(self, p):
        """
        :param Point p:
        """
        return Point(self.c + p.c)

    def __iadd__(self, p):
        """
        :param Point p:
        """
        self.c += p.c
        return self

    def __sub__(self, p):
        """
        :param Point p:
        """
        return Point(self.c - p.c)

    def __isub__(self, p):
        """
        :param Point p:
        """
        self.c -= p.c
        return self

    def __mul__(self, f: float):
        return Point(self.c * f)

    def __imul__(self, f: float):
        self.c *= f
        return self

    def __truediv__(self, f: float):
        return Point(self.c / f)

    def __itruediv__(self, f: float):
        self.c /= f
        return self

    def __repr__(self):
        return "({}, {})".format(round(self.x, 10), round(self.y, 10))

    def __neg__(self):
        return Point(-self.c)

    def __eq__(self, p):
        return abs(self.c - p.c) < self.EPS

    def __abs__(self):
        return abs(self.c)

    def dot(self, p):
        """
        内積
        :param Point p:
        :rtype: float
        """
        return self.x * p.x + self.y * p.y

    def det(self, p):
        """
        外積
        :param Point p:
        :rtype: float
        """
        return self.x * p.y - self.y * p.x

    def dist(self, p):
        """
        距離
        :param Point p:
        :rtype: float
        """
        return abs(self.c - p.c)

    def r(self):
        """
        原点からの距離
        :rtype: float
        """
        return abs(self.c)

    def phase(self):
        """
        原点からの角度
        :rtype: float
        """
        return cmath.phase(self.c)

    def angle(self, p, q):
        """
        p に向かってる状態から q まで反時計回りに回転するときの角度
        :param Point p:
        :param Point q:
        :rtype: float
        """
        return (cmath.phase(q.c - self.c) - cmath.phase(p.c - self.c)) % TAU

    def area(self, p, q):
        """
        p, q となす三角形の面積
        :param Point p:
        :param Point q:
        :rtype: float
        """
        return abs((p - self).det(q - self) / 2)

    def projection_point(self, p, q, allow_outer=False):
        """
        線分 pq を通る直線上に垂線をおろしたときの足の座標
        :param Point p:
        :param Point q:
        :param allow_outer: 答えが線分の間になくても OK
        :rtype: Point|None
        """
        diff_q = q - p
        
        r = (self - p).dot(diff_q) / abs(diff_q)
        
        phase = diff_q.phase()

        ret = Point.from_polar(r, phase) + p
        if allow_outer or (p - ret).dot(q - ret) < self.EPS:
            return ret
        return None

    def on_segment(self, p, q):
        """
        点が線分 pq の上に乗っているか
        :param Point p:
        :param Point q:
        :rtype: bool
        """
        return abs((p - self).det(q - self)) < self.EPS and (p - self).dot(q - self) < self.EPS


p1x, p1y, p2x, p2y = list(map(int, sys.stdin.buffer.readline().split()))
Q = int(sys.stdin.buffer.readline())
XY = [list(map(int, sys.stdin.buffer.readline().split())) for _ in range(Q)]

p1 = Point.from_rect(p1x, p1y)
p2 = Point.from_rect(p2x, p2y)

for x, y in XY:
    p = Point.from_rect(x, y)
    ans = p.projection_point(p1, p2, allow_outer=True)
    print('{:.10f} {:.10f}'.format(ans.x, ans.y))

Solution (Python):


import array
from fractions import Fraction
import math
import os
import sys


class Vec(object):
    def __init__(self, x, y):
        self.x = x
        self.y = y
        super().__init__()

    def __add__(self, other):
        return Vec(self.x + other.x, self.y + other.y)

    def __sub__(self, other):
        return Vec(self.x - other.x, self.y - other.y)

    def __mul__(self, scalar):
        return Vec(self.x * scalar, self.y * scalar)

    def __truediv__(self, scalar):
        return Vec(self.x / scalar, self.y / scalar)

    def __iadd__(self, other):
        self.x += other.x
        self.y += other.y
        return self

    def __isub__(self, other):
        self.x -= other.x
        self.y -= other.y
        return self

    def __imul__(self, scalar):
        self.x *= scalar
        self.y *= scalar
        return self

    def __idiv__(self, scalar):
        self.x /= scalar
        self.y /= scalar
        return self

    def __neg__(self):
        return Vec(-self.x, -self.y)

    def dot(self, other):
        return self.x * other.x + self.y * other.y

    def cross(self, other):
        return self.x * other.y - self.y * other.x

    def abs2(self):
        return self.x * self.x + self.y * self.y

    def __abs__(self):
        return math.sqrt(float(self.abs2()))

    def __str__(self):
        return '({}, {})'.format(self.x, self.y)


def main():
    x1, y1, x2, y2 = read_ints()
    Q = read_int()
    for _ in range(Q):
        x, y = read_ints()
        print(*solve(Vec(x1, y1), Vec(x2, y2), Vec(x, y)))


def solve(u, v, a):
    v -= u
    a -= u
    f = Fraction(v.dot(a), v.abs2())
    v *= f
    v += u
    return float(v.x), float(v.y)





DEBUG = 'DEBUG' in os.environ


def inp():
    return sys.stdin.readline().rstrip()


def read_int():
    return int(inp())


def read_ints():
    return [int(e) for e in inp().split()]


def dprint(*value, sep=' ', end='\n'):
    if DEBUG:
        print(*value, sep=sep, end=end)


if __name__ == '__main__':
    main()

Solution (Python):
import sys
input = sys.stdin.readline

def main():
    x1, y1, x2, y2 = list(map(int, input().split()))
    a = [x2 - x1, y2 - y1]
    lal2 = a[0]**2 + a[1]**2
    q = int(input())
    for i in range(q):
        x, y = map(int, input().split())
        b = [x - x1, y - y1]
        t = (a[0]*b[0] + a[1]*b[1]) / lal2
        print(x1 + a[0] * t, y1 + a[1] * t)

if __name__ == "__main__":
    main()

Solution (Python):


import sys

sys.setrecursionlimit(10 ** 9)
def input(): return sys.stdin.readline().strip()
def INT(): return int(input())
def MAP(): return map(int, input().split())
def LIST(): return list(map(int, input().split()))
INF=float('inf')

def dot(a, b):
    x1, y1 = a
    x2, y2 = b
    return x1*x2 + y1*y2

def norm(a):
    x, y = a
    return x**2 + y**2

def add(a, b):
    x1, y1 = a
    x2, y2 = b
    return (x1+x2, y1+y2)

def sub(a, b):
    x1, y1 = a
    x2, y2 = b
    return (x1-x2, y1-y2)

def mul(a, b):
    x1, y1 = a
    if not isinstance(b, tuple):
        return (x1*b, y1*b)
    x2, y2 = b 
    return (x1*x2, y1*y2)


def project(seg, p):
    p1, p2 = seg
    base = sub(p2, p1)
    r = dot(sub(p, p1), base) / norm(base)
    return add(p1, mul(base, r))

x1, y1, x2, y2 = MAP()
seg = ((x1, y1), (x2, y2))
Q = INT()
for i in range(Q):
    x, y  = MAP()
    res = project(seg, (x, y))
    print(*res)

Solution (Python):
if __name__ == "__main__":
    x_p1, y_p1, x_p2, y_p2 = map(lambda x: int(x), input().split())
    q = int(input())

    p1p2 = (x_p2 - x_p1, y_p2 - y_p1)
    for _ in range(q):
        x_p0, y_p0 = map(lambda x: int(x), input().split())
        p1p0 = (x_p0 - x_p1, y_p0 - y_p1)
        r = (p1p2[0] * p1p0[0] + p1p2[1] * p1p0[1]) / (p1p2[0] ** 2 + p1p2[1] ** 2)
        print(x_p1 + p1p2[0] * r, y_p1 + p1p2[1] * r)

Solution (Python):
import math
class Point:
	def __init__(self,x,y):
		self.x = x
		self.y = y
	def __add__(self,p):
		return Point(self.x+p.x,self.y+p.y)
	def __sub__(self,p):
		return Point(self.x-p.x,self.y-p.y)
	def __mul__(self,p):
		return Point(self.x*p,self.y*p)

def Dot(a,b):
	return a.x*b.x + a.y*b.y
def Cross(a,b):
	return a.x*b.y - a.y*b.x
def Dis(a,b):
	return math.sqrt( (a.x-b.x)*(a.x-b.x) + (a.y-b.y)*(a.y-b.y) )

x1,y1,x2,y2 = map(int,input().split())
p1 = Point(x1,y1)
p2 = Point(x2,y2)
p1p2_dis = Dis(p1,p2)

q = int(input())
for i in range(q):
	x,y = map(int,input().split())
	p = Point(x,y)
	dot_res = Dot(p-p1,p2-p1)
	len = dot_res / p1p2_dis
	x = (p2-p1) * (len/p1p2_dis) + p1
	print(x.x,x.y)

Solution (Python):
def Projection(point, begin, end):
	tmp = (point-begin) / (end-begin)
	return tmp.real * (end-begin) + begin

a, b, c, d = map(int, input().split())
q, r = complex(a, b), complex(c, d)
n = int(input())
for _ in range(n):
	x, y = map(int, input().split())
	p = complex(x, y)
	proj = Projection(p, q, r)
	print(proj.real, proj.imag)

Solution (Python):
L1=input().split( )
[xp1,yp1,xp2,yp2]=[int(L1[0]),int(L1[1]),int(L1[2]),int(L1[3])]
direction=[xp2-xp1,yp2-yp1]
norm_direc=[(xp2-xp1)/((xp2-xp1)**2+(yp2-yp1)**2)**(1/2),(yp2-yp1)/((xp2-xp1)**2+(yp2-yp1)**2)**(1/2)]

q=int(input())
L=[]
for i in range(q):
    L2=input().split( )
    L.append([int(L2[0]),int(L2[1])])

Point=[]
for i in range(q):
    Point.append([(L[i][0]-xp1),(L[i][1]-yp1)])

Vec=[]
for i in range(q):
    Vec.append([L[i][0]-xp1,L[i][1]-yp1])
Proj=[]
for i in range(q):
    inter=Vec[i][0]*norm_direc[0]+Vec[i][1]*norm_direc[1]
    Proj.append([norm_direc[0]*inter,norm_direc[1]*inter])

L2=[]

for i in range(q):
    L2.append([xp1+Proj[i][0],yp1+Proj[i][1]])
for i in range(q):
    print(*L2[i])