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How to check if a given point lies inside or outside a polygon?
• Difficulty Level : Hard
• Last Updated : 11 Dec, 2020

Given a polygon and a point ‘p’, find if ‘p’ lies inside the polygon or not. The points lying on the border are considered inside. We strongly recommend to see the following post first.
How to check if two given line segments intersect?
Following is a simple idea to check whether a point is inside or outside.

```1) Draw a horizontal line to the right of each point and extend it to infinity

1) Count the number of times the line intersects with polygon edges.

2) A point is inside the polygon if either count of intersections is odd or
point lies on an edge of polygon.  If none of the conditions is true, then
point lies outside.``` How to handle point ‘g’ in the above figure?
Note that we should return true if the point lies on the line or same as one of the vertices of the given polygon. To handle this, after checking if the line from ‘p’ to extreme intersects, we check whether ‘p’ is colinear with vertices of current line of polygon. If it is coliear, then we check if the point ‘p’ lies on current side of polygon, if it lies, we return true, else false.

Following is the implementation of the above idea.

## C++

 `// A C++ program to check if a given point lies inside a given polygon``// Refer https://www.geeksforgeeks.org/check-if-two-given-line-segments-intersect/``// for explanation of functions onSegment(), orientation() and doIntersect()``#include ``using` `namespace` `std;` `// Define Infinite (Using INT_MAX caused overflow problems)``#define INF 10000` `struct` `Point``{``    ``int` `x;``    ``int` `y;``};` `// Given three colinear points p, q, r, the function checks if``// point q lies on line segment 'pr'``bool` `onSegment(Point p, Point q, Point r)``{``    ``if` `(q.x <= max(p.x, r.x) && q.x >= min(p.x, r.x) &&``            ``q.y <= max(p.y, r.y) && q.y >= min(p.y, r.y))``        ``return` `true``;``    ``return` `false``;``}` `// To find orientation of ordered triplet (p, q, r).``// The function returns following values``// 0 --> p, q and r are colinear``// 1 --> Clockwise``// 2 --> Counterclockwise``int` `orientation(Point p, Point q, Point r)``{``    ``int` `val = (q.y - p.y) * (r.x - q.x) -``            ``(q.x - p.x) * (r.y - q.y);` `    ``if` `(val == 0) ``return` `0; ``// colinear``    ``return` `(val > 0)? 1: 2; ``// clock or counterclock wise``}` `// The function that returns true if line segment 'p1q1'``// and 'p2q2' intersect.``bool` `doIntersect(Point p1, Point q1, Point p2, Point q2)``{``    ``// Find the four orientations needed for general and``    ``// special cases``    ``int` `o1 = orientation(p1, q1, p2);``    ``int` `o2 = orientation(p1, q1, q2);``    ``int` `o3 = orientation(p2, q2, p1);``    ``int` `o4 = orientation(p2, q2, q1);` `    ``// General case``    ``if` `(o1 != o2 && o3 != o4)``        ``return` `true``;` `    ``// Special Cases``    ``// p1, q1 and p2 are colinear and p2 lies on segment p1q1``    ``if` `(o1 == 0 && onSegment(p1, p2, q1)) ``return` `true``;` `    ``// p1, q1 and p2 are colinear and q2 lies on segment p1q1``    ``if` `(o2 == 0 && onSegment(p1, q2, q1)) ``return` `true``;` `    ``// p2, q2 and p1 are colinear and p1 lies on segment p2q2``    ``if` `(o3 == 0 && onSegment(p2, p1, q2)) ``return` `true``;` `    ``// p2, q2 and q1 are colinear and q1 lies on segment p2q2``    ``if` `(o4 == 0 && onSegment(p2, q1, q2)) ``return` `true``;` `    ``return` `false``; ``// Doesn't fall in any of the above cases``}` `// Returns true if the point p lies inside the polygon[] with n vertices``bool` `isInside(Point polygon[], ``int` `n, Point p)``{``    ``// There must be at least 3 vertices in polygon[]``    ``if` `(n < 3) ``return` `false``;` `    ``// Create a point for line segment from p to infinite``    ``Point extreme = {INF, p.y};` `    ``// Count intersections of the above line with sides of polygon``    ``int` `count = 0, i = 0;``    ``do``    ``{``        ``int` `next = (i+1)%n;` `        ``// Check if the line segment from 'p' to 'extreme' intersects``        ``// with the line segment from 'polygon[i]' to 'polygon[next]'``        ``if` `(doIntersect(polygon[i], polygon[next], p, extreme))``        ``{``            ``// If the point 'p' is colinear with line segment 'i-next',``            ``// then check if it lies on segment. If it lies, return true,``            ``// otherwise false``            ``if` `(orientation(polygon[i], p, polygon[next]) == 0)``            ``return` `onSegment(polygon[i], p, polygon[next]);` `            ``count++;``        ``}``        ``i = next;``    ``} ``while` `(i != 0);` `    ``// Return true if count is odd, false otherwise``    ``return` `count&1; ``// Same as (count%2 == 1)``}` `// Driver program to test above functions``int` `main()``{``    ``Point polygon1[] = {{0, 0}, {10, 0}, {10, 10}, {0, 10}};``    ``int` `n = ``sizeof``(polygon1)/``sizeof``(polygon1);``    ``Point p = {20, 20};``    ``isInside(polygon1, n, p)? cout << ``"Yes \n"``: cout << ``"No \n"``;` `    ``p = {5, 5};``    ``isInside(polygon1, n, p)? cout << ``"Yes \n"``: cout << ``"No \n"``;` `    ``Point polygon2[] = {{0, 0}, {5, 5}, {5, 0}};``    ``p = {3, 3};``    ``n = ``sizeof``(polygon2)/``sizeof``(polygon2);``    ``isInside(polygon2, n, p)? cout << ``"Yes \n"``: cout << ``"No \n"``;` `    ``p = {5, 1};``    ``isInside(polygon2, n, p)? cout << ``"Yes \n"``: cout << ``"No \n"``;` `    ``p = {8, 1};``    ``isInside(polygon2, n, p)? cout << ``"Yes \n"``: cout << ``"No \n"``;` `    ``Point polygon3[] = {{0, 0}, {10, 0}, {10, 10}, {0, 10}};``    ``p = {-1,10};``    ``n = ``sizeof``(polygon3)/``sizeof``(polygon3);``    ``isInside(polygon3, n, p)? cout << ``"Yes \n"``: cout << ``"No \n"``;` `    ``return` `0;``}`

## Java

 `// A Java program to check if a given point``// lies inside a given polygon``// Refer https://www.geeksforgeeks.org/check-if-two-given-line-segments-intersect/``// for explanation of functions onSegment(),``// orientation() and doIntersect()``class` `GFG``{` `    ``// Define Infinite (Using INT_MAX``    ``// caused overflow problems)``    ``static` `int` `INF = ``10000``;` `    ``static` `class` `Point``    ``{``        ``int` `x;``        ``int` `y;` `        ``public` `Point(``int` `x, ``int` `y)``        ``{``            ``this``.x = x;``            ``this``.y = y;``        ``}``    ``};` `    ``// Given three colinear points p, q, r,``    ``// the function checks if point q lies``    ``// on line segment 'pr'``    ``static` `boolean` `onSegment(Point p, Point q, Point r)``    ``{``        ``if` `(q.x <= Math.max(p.x, r.x) &&``            ``q.x >= Math.min(p.x, r.x) &&``            ``q.y <= Math.max(p.y, r.y) &&``            ``q.y >= Math.min(p.y, r.y))``        ``{``            ``return` `true``;``        ``}``        ``return` `false``;``    ``}` `    ``// To find orientation of ordered triplet (p, q, r).``    ``// The function returns following values``    ``// 0 --> p, q and r are colinear``    ``// 1 --> Clockwise``    ``// 2 --> Counterclockwise``    ``static` `int` `orientation(Point p, Point q, Point r)``    ``{``        ``int` `val = (q.y - p.y) * (r.x - q.x)``                ``- (q.x - p.x) * (r.y - q.y);` `        ``if` `(val == ``0``)``        ``{``            ``return` `0``; ``// colinear``        ``}``        ``return` `(val > ``0``) ? ``1` `: ``2``; ``// clock or counterclock wise``    ``}` `    ``// The function that returns true if``    ``// line segment 'p1q1' and 'p2q2' intersect.``    ``static` `boolean` `doIntersect(Point p1, Point q1,``                            ``Point p2, Point q2)``    ``{``        ``// Find the four orientations needed for``        ``// general and special cases``        ``int` `o1 = orientation(p1, q1, p2);``        ``int` `o2 = orientation(p1, q1, q2);``        ``int` `o3 = orientation(p2, q2, p1);``        ``int` `o4 = orientation(p2, q2, q1);` `        ``// General case``        ``if` `(o1 != o2 && o3 != o4)``        ``{``            ``return` `true``;``        ``}` `        ``// Special Cases``        ``// p1, q1 and p2 are colinear and``        ``// p2 lies on segment p1q1``        ``if` `(o1 == ``0` `&& onSegment(p1, p2, q1))``        ``{``            ``return` `true``;``        ``}` `        ``// p1, q1 and p2 are colinear and``        ``// q2 lies on segment p1q1``        ``if` `(o2 == ``0` `&& onSegment(p1, q2, q1))``        ``{``            ``return` `true``;``        ``}` `        ``// p2, q2 and p1 are colinear and``        ``// p1 lies on segment p2q2``        ``if` `(o3 == ``0` `&& onSegment(p2, p1, q2))``        ``{``            ``return` `true``;``        ``}` `        ``// p2, q2 and q1 are colinear and``        ``// q1 lies on segment p2q2``        ``if` `(o4 == ``0` `&& onSegment(p2, q1, q2))``        ``{``            ``return` `true``;``        ``}` `        ``// Doesn't fall in any of the above cases``        ``return` `false``;``    ``}` `    ``// Returns true if the point p lies``    ``// inside the polygon[] with n vertices``    ``static` `boolean` `isInside(Point polygon[], ``int` `n, Point p)``    ``{``        ``// There must be at least 3 vertices in polygon[]``        ``if` `(n < ``3``)``        ``{``            ``return` `false``;``        ``}` `        ``// Create a point for line segment from p to infinite``        ``Point extreme = ``new` `Point(INF, p.y);` `        ``// Count intersections of the above line``        ``// with sides of polygon``        ``int` `count = ``0``, i = ``0``;``        ``do``        ``{``            ``int` `next = (i + ``1``) % n;` `            ``// Check if the line segment from 'p' to``            ``// 'extreme' intersects with the line``            ``// segment from 'polygon[i]' to 'polygon[next]'``            ``if` `(doIntersect(polygon[i], polygon[next], p, extreme))``            ``{``                ``// If the point 'p' is colinear with line``                ``// segment 'i-next', then check if it lies``                ``// on segment. If it lies, return true, otherwise false``                ``if` `(orientation(polygon[i], p, polygon[next]) == ``0``)``                ``{``                    ``return` `onSegment(polygon[i], p,``                                    ``polygon[next]);``                ``}` `                ``count++;``            ``}``            ``i = next;``        ``} ``while` `(i != ``0``);` `        ``// Return true if count is odd, false otherwise``        ``return` `(count % ``2` `== ``1``); ``// Same as (count%2 == 1)``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String[] args)``    ``{``        ``Point polygon1[] = {``new` `Point(``0``, ``0``),``                            ``new` `Point(``10``, ``0``),``                            ``new` `Point(``10``, ``10``),``                            ``new` `Point(``0``, ``10``)};``        ``int` `n = polygon1.length;``        ``Point p = ``new` `Point(``20``, ``20``);``        ``if` `(isInside(polygon1, n, p))``        ``{``            ``System.out.println(``"Yes"``);``        ``}``        ``else``        ``{``            ``System.out.println(``"No"``);``        ``}``        ``p = ``new` `Point(``5``, ``5``);``        ``if` `(isInside(polygon1, n, p))``        ``{``            ``System.out.println(``"Yes"``);``        ``}``        ``else``        ``{``            ``System.out.println(``"No"``);``        ``}``        ``Point polygon2[] = {``new` `Point(``0``, ``0``),``            ``new` `Point(``5``, ``5``), ``new` `Point(``5``, ``0``)};``        ``p = ``new` `Point(``3``, ``3``);``        ``n = polygon2.length;``        ``if` `(isInside(polygon2, n, p))``        ``{``            ``System.out.println(``"Yes"``);``        ``}``        ``else``        ``{``            ``System.out.println(``"No"``);``        ``}``        ``p = ``new` `Point(``5``, ``1``);``        ``if` `(isInside(polygon2, n, p))``        ``{``            ``System.out.println(``"Yes"``);``        ``}``        ``else``        ``{``            ``System.out.println(``"No"``);``        ``}``        ``p = ``new` `Point(``8``, ``1``);``        ``if` `(isInside(polygon2, n, p))``        ``{``            ``System.out.println(``"Yes"``);``        ``}``        ``else``        ``{``            ``System.out.println(``"No"``);``        ``}``        ``Point polygon3[] = {``new` `Point(``0``, ``0``),``                            ``new` `Point(``10``, ``0``),``                            ``new` `Point(``10``, ``10``),``                            ``new` `Point(``0``, ``10``)};``        ``p = ``new` `Point(-``1``, ``10``);``        ``n = polygon3.length;``        ``if` `(isInside(polygon3, n, p))``        ``{``            ``System.out.println(``"Yes"``);``        ``}``        ``else``        ``{``            ``System.out.println(``"No"``);``        ``}``    ``}``}` `// This code is contributed by 29AjayKumar`

## Python3

 `# A Python3 program to check if a given point ``# lies inside a given polygon``# Refer https://www.geeksforgeeks.org/check-if-two-given-line-segments-intersect/``# for explanation of functions onSegment(),``# orientation() and doIntersect() ` `# Define Infinite (Using INT_MAX ``# caused overflow problems)``INT_MAX ``=` `10000` `# Given three colinear points p, q, r, ``# the function checks if point q lies``# on line segment 'pr'``def` `onSegment(p:``tuple``, q:``tuple``, r:``tuple``) ``-``> ``bool``:``    ` `    ``if` `((q[``0``] <``=` `max``(p[``0``], r[``0``])) &``        ``(q[``0``] >``=` `min``(p[``0``], r[``0``])) &``        ``(q[``1``] <``=` `max``(p[``1``], r[``1``])) &``        ``(q[``1``] >``=` `min``(p[``1``], r[``1``]))):``        ``return` `True``        ` `    ``return` `False` `# To find orientation of ordered triplet (p, q, r).``# The function returns following values``# 0 --> p, q and r are colinear``# 1 --> Clockwise``# 2 --> Counterclockwise``def` `orientation(p:``tuple``, q:``tuple``, r:``tuple``) ``-``> ``int``:``    ` `    ``val ``=` `(((q[``1``] ``-` `p[``1``]) ``*``            ``(r[``0``] ``-` `q[``0``])) ``-``           ``((q[``0``] ``-` `p[``0``]) ``*``            ``(r[``1``] ``-` `q[``1``])))``           ` `    ``if` `val ``=``=` `0``:``        ``return` `0``    ``if` `val > ``0``:``        ``return` `1` `# Collinear``    ``else``:``        ``return` `2` `# Clock or counterclock` `def` `doIntersect(p1, q1, p2, q2):``    ` `    ``# Find the four orientations needed for ``    ``# general and special cases``    ``o1 ``=` `orientation(p1, q1, p2)``    ``o2 ``=` `orientation(p1, q1, q2)``    ``o3 ``=` `orientation(p2, q2, p1)``    ``o4 ``=` `orientation(p2, q2, q1)` `    ``# General case``    ``if` `(o1 !``=` `o2) ``and` `(o3 !``=` `o4):``        ``return` `True``    ` `    ``# Special Cases``    ``# p1, q1 and p2 are colinear and``    ``# p2 lies on segment p1q1``    ``if` `(o1 ``=``=` `0``) ``and` `(onSegment(p1, p2, q1)):``        ``return` `True` `    ``# p1, q1 and p2 are colinear and``    ``# q2 lies on segment p1q1``    ``if` `(o2 ``=``=` `0``) ``and` `(onSegment(p1, q2, q1)):``        ``return` `True` `    ``# p2, q2 and p1 are colinear and``    ``# p1 lies on segment p2q2``    ``if` `(o3 ``=``=` `0``) ``and` `(onSegment(p2, p1, q2)):``        ``return` `True` `    ``# p2, q2 and q1 are colinear and``    ``# q1 lies on segment p2q2``    ``if` `(o4 ``=``=` `0``) ``and` `(onSegment(p2, q1, q2)):``        ``return` `True` `    ``return` `False` `# Returns true if the point p lies ``# inside the polygon[] with n vertices``def` `is_inside_polygon(points:``list``, p:``tuple``) ``-``> ``bool``:``    ` `    ``n ``=` `len``(points)``    ` `    ``# There must be at least 3 vertices``    ``# in polygon``    ``if` `n < ``3``:``        ``return` `False``        ` `    ``# Create a point for line segment``    ``# from p to infinite``    ``extreme ``=` `(INT_MAX, p[``1``])``    ``count ``=` `i ``=` `0``    ` `    ``while` `True``:``        ``next` `=` `(i ``+` `1``) ``%` `n``        ` `        ``# Check if the line segment from 'p' to ``        ``# 'extreme' intersects with the line ``        ``# segment from 'polygon[i]' to 'polygon[next]'``        ``if` `(doIntersect(points[i],``                        ``points[``next``],``                        ``p, extreme)):``                            ` `            ``# If the point 'p' is colinear with line ``            ``# segment 'i-next', then check if it lies ``            ``# on segment. If it lies, return true, otherwise false``            ``if` `orientation(points[i], p,``                           ``points[``next``]) ``=``=` `0``:``                ``return` `onSegment(points[i], p,``                                 ``points[``next``])``                                 ` `            ``count ``+``=` `1``            ` `        ``i ``=` `next``        ` `        ``if` `(i ``=``=` `0``):``            ``break``        ` `    ``# Return true if count is odd, false otherwise``    ``return` `(count ``%` `2` `=``=` `1``)` `# Driver code``if` `__name__ ``=``=` `'__main__'``:``    ` `    ``polygon1 ``=` `[ (``0``, ``0``), (``10``, ``0``), (``10``, ``10``), (``0``, ``10``) ]``    ` `    ``p ``=` `(``20``, ``20``)``    ``if` `(is_inside_polygon(points ``=` `polygon1, p ``=` `p)):``      ``print` `(``'Yes'``)``    ``else``:``      ``print` `(``'No'``)``      ` `    ``p ``=` `(``5``, ``5``)``    ``if` `(is_inside_polygon(points ``=` `polygon1, p ``=` `p)):``      ``print` `(``'Yes'``)``    ``else``:``      ``print` `(``'No'``)` `    ``polygon2 ``=` `[ (``0``, ``0``), (``5``, ``0``), (``5``, ``5``), (``3``, ``3``) ]``    ` `    ``p ``=` `(``3``, ``3``)``    ``if` `(is_inside_polygon(points ``=` `polygon2, p ``=` `p)):``      ``print` `(``'Yes'``)``    ``else``:``      ``print` `(``'No'``)``      ` `    ``p ``=` `(``5``, ``1``)``    ``if` `(is_inside_polygon(points ``=` `polygon2, p ``=` `p)):``      ``print` `(``'Yes'``)``    ``else``:``      ``print` `(``'No'``)``      ` `    ``p ``=` `(``8``, ``1``)``    ``if` `(is_inside_polygon(points ``=` `polygon2, p ``=` `p)):``      ``print` `(``'Yes'``)``    ``else``:``      ``print` `(``'No'``)``    ` `    ``polygon3 ``=` `[ (``0``, ``0``), (``10``, ``0``), (``10``, ``10``), (``0``, ``10``) ]``    ` `    ``p ``=` `(``-``1``, ``10``)``    ``if` `(is_inside_polygon(points ``=` `polygon3, p ``=` `p)):``      ``print` `(``'Yes'``)``    ``else``:``      ``print` `(``'No'``)``      ` `# This code is contributed by Vikas Chitturi`

## C#

 `// A C# program to check if a given point``// lies inside a given polygon``// Refer https://www.geeksforgeeks.org/check-if-two-given-line-segments-intersect/``// for explanation of functions onSegment(),``// orientation() and doIntersect()``using` `System;` `class` `GFG``{` `    ``// Define Infinite (Using INT_MAX``    ``// caused overflow problems)``    ``static` `int` `INF = 10000;` `    ``class` `Point``    ``{``        ``public` `int` `x;``        ``public` `int` `y;` `        ``public` `Point(``int` `x, ``int` `y)``        ``{``            ``this``.x = x;``            ``this``.y = y;``        ``}``    ``};` `    ``// Given three colinear points p, q, r,``    ``// the function checks if point q lies``    ``// on line segment 'pr'``    ``static` `bool` `onSegment(Point p, Point q, Point r)``    ``{``        ``if` `(q.x <= Math.Max(p.x, r.x) &&``            ``q.x >= Math.Min(p.x, r.x) &&``            ``q.y <= Math.Max(p.y, r.y) &&``            ``q.y >= Math.Min(p.y, r.y))``        ``{``            ``return` `true``;``        ``}``        ``return` `false``;``    ``}` `    ``// To find orientation of ordered triplet (p, q, r).``    ``// The function returns following values``    ``// 0 --> p, q and r are colinear``    ``// 1 --> Clockwise``    ``// 2 --> Counterclockwise``    ``static` `int` `orientation(Point p, Point q, Point r)``    ``{``        ``int` `val = (q.y - p.y) * (r.x - q.x) -``                ``(q.x - p.x) * (r.y - q.y);` `        ``if` `(val == 0)``        ``{``            ``return` `0; ``// colinear``        ``}``        ``return` `(val > 0) ? 1 : 2; ``// clock or counterclock wise``    ``}` `    ``// The function that returns true if``    ``// line segment 'p1q1' and 'p2q2' intersect.``    ``static` `bool` `doIntersect(Point p1, Point q1,``                            ``Point p2, Point q2)``    ``{``        ``// Find the four orientations needed for``        ``// general and special cases``        ``int` `o1 = orientation(p1, q1, p2);``        ``int` `o2 = orientation(p1, q1, q2);``        ``int` `o3 = orientation(p2, q2, p1);``        ``int` `o4 = orientation(p2, q2, q1);` `        ``// General case``        ``if` `(o1 != o2 && o3 != o4)``        ``{``            ``return` `true``;``        ``}` `        ``// Special Cases``        ``// p1, q1 and p2 are colinear and``        ``// p2 lies on segment p1q1``        ``if` `(o1 == 0 && onSegment(p1, p2, q1))``        ``{``            ``return` `true``;``        ``}` `        ``// p1, q1 and p2 are colinear and``        ``// q2 lies on segment p1q1``        ``if` `(o2 == 0 && onSegment(p1, q2, q1))``        ``{``            ``return` `true``;``        ``}` `        ``// p2, q2 and p1 are colinear and``        ``// p1 lies on segment p2q2``        ``if` `(o3 == 0 && onSegment(p2, p1, q2))``        ``{``            ``return` `true``;``        ``}` `        ``// p2, q2 and q1 are colinear and``        ``// q1 lies on segment p2q2``        ``if` `(o4 == 0 && onSegment(p2, q1, q2))``        ``{``            ``return` `true``;``        ``}` `        ``// Doesn't fall in any of the above cases``        ``return` `false``;``    ``}` `    ``// Returns true if the point p lies``    ``// inside the polygon[] with n vertices``    ``static` `bool` `isInside(Point []polygon, ``int` `n, Point p)``    ``{``        ``// There must be at least 3 vertices in polygon[]``        ``if` `(n < 3)``        ``{``            ``return` `false``;``        ``}` `        ``// Create a point for line segment from p to infinite``        ``Point extreme = ``new` `Point(INF, p.y);` `        ``// Count intersections of the above line``        ``// with sides of polygon``        ``int` `count = 0, i = 0;``        ``do``        ``{``            ``int` `next = (i + 1) % n;` `            ``// Check if the line segment from 'p' to``            ``// 'extreme' intersects with the line``            ``// segment from 'polygon[i]' to 'polygon[next]'``            ``if` `(doIntersect(polygon[i],``                            ``polygon[next], p, extreme))``            ``{``                ``// If the point 'p' is colinear with line``                ``// segment 'i-next', then check if it lies``                ``// on segment. If it lies, return true, otherwise false``                ``if` `(orientation(polygon[i], p, polygon[next]) == 0)``                ``{``                    ``return` `onSegment(polygon[i], p,``                                    ``polygon[next]);``                ``}``                ``count++;``            ``}``            ``i = next;``        ``} ``while` `(i != 0);` `        ``// Return true if count is odd, false otherwise``        ``return` `(count % 2 == 1); ``// Same as (count%2 == 1)``    ``}` `    ``// Driver Code``    ``public` `static` `void` `Main(String[] args)``    ``{``        ``Point []polygon1 = {``new` `Point(0, 0),``                            ``new` `Point(10, 0),``                            ``new` `Point(10, 10),``                            ``new` `Point(0, 10)};``        ``int` `n = polygon1.Length;``        ``Point p = ``new` `Point(20, 20);``        ``if` `(isInside(polygon1, n, p))``        ``{``            ``Console.WriteLine(``"Yes"``);``        ``}``        ``else``        ``{``            ``Console.WriteLine(``"No"``);``        ``}``        ``p = ``new` `Point(5, 5);``        ``if` `(isInside(polygon1, n, p))``        ``{``            ``Console.WriteLine(``"Yes"``);``        ``}``        ``else``        ``{``            ``Console.WriteLine(``"No"``);``        ``}``        ``Point []polygon2 = {``new` `Point(0, 0),``                            ``new` `Point(5, 5),``                            ``new` `Point(5, 0)};``        ``p = ``new` `Point(3, 3);``        ``n = polygon2.Length;``        ``if` `(isInside(polygon2, n, p))``        ``{``            ``Console.WriteLine(``"Yes"``);``        ``}``        ``else``        ``{``            ``Console.WriteLine(``"No"``);``        ``}``        ``p = ``new` `Point(5, 1);``        ``if` `(isInside(polygon2, n, p))``        ``{``            ``Console.WriteLine(``"Yes"``);``        ``}``        ``else``        ``{``            ``Console.WriteLine(``"No"``);``        ``}``        ``p = ``new` `Point(8, 1);``        ``if` `(isInside(polygon2, n, p))``        ``{``            ``Console.WriteLine(``"Yes"``);``        ``}``        ``else``        ``{``            ``Console.WriteLine(``"No"``);``        ``}``        ``Point []polygon3 = {``new` `Point(0, 0),``                            ``new` `Point(10, 0),``                            ``new` `Point(10, 10),``                            ``new` `Point(0, 10)};``        ``p = ``new` `Point(-1, 10);``        ``n = polygon3.Length;``        ``if` `(isInside(polygon3, n, p))``        ``{``            ``Console.WriteLine(``"Yes"``);``        ``}``        ``else``        ``{``            ``Console.WriteLine(``"No"``);``        ``}``    ``}``}` `// This code is contributed by 29AjayKumar`

Output:

```No
Yes
Yes
Yes
No
No```

Time Complexity: O(n) where n is the number of vertices in the given polygon.