## Amazon Interview Question

Software Engineer / Developers**Country:**India

**Interview Type:**In-Person

This will fail if P is located outside the triangle in some specific cases. Consider the triangle made of coordinates - (3,0)A ; (3,2)B ; and (0,2)C. Now, P(0,0) is outside the triangle ABC, but still the areas are a match, You need an extra check to see of the point P is inside the triangle; such as explained in the post below - about P being in the same side as one of A, B or C - when a line by other two points splits the plane

For each point, test if the origin is on the line or on the same side as the current point when the plane is split by the line that the other two points make.

```
from collections import namedtuple
Point = namedtuple('Point', 'x y')
ORIGIN = Point(0, 0)
def contains_origin(points):
for i in xrange(len(points)):
p = points[i]
rest = points[:i] + points[i + 1:]
origin_position = above_or_below_or_on(ORIGIN, rest)
if origin_position == 'on':
return True
elif origin_position != above_or_below_or_on(p, rest):
return False
return True
def above_or_below_or_on(point, rest):
a = rest[0]
b = rest[1]
d = float(a.y - b.y) / (a.x - b.x) * (point.x - b.x) + b.y
if point.y > d:
return 'above'
elif point.y < d:
return 'below'
else:
return 'on'
if __name__ == '__main__':
print(contains_origin([
Point(1, 0),
Point(0, 3),
Point(-2, -1),
]))
```

Seems the easiest way to do this would be to do some kind of radial sweep from the origin. Then moving from p1 to p2 and p3 you simply add up the angle between p1 and p2 then p2 and p3. If that angel is greater than 180 degrees then the origin must be in the triangle. If it equaled 180 then you must be on an edge. You would also immediately fail if you found 2 points on the same sweep line.

Sort the x-coordinates by considering all three points and get {x1, x2, x3} in sorted order.

Sort the y-coordinates by considering all three points and get {y1, y2, y3} in sorted order.

If for given point P(a, b), if (x1<= a <=x3) && (y1<=b<=y3), then the point lies within the triangle, else it lies outside

Let the three points be

p1 = (x1, y1)

p2 = (x2, y2)

p3 = (x3, y3)

and the points are sorted in increasing order of x values, i.e. x1 <= x2 <= x3.

If the origin (0, 0) is inside the triangle then it must be the case that either

case 1:

x1 <= 0 <= x2

now use linear interpolation to obtain an equation y12(x), and y13(x). Where these are equations of line that pass through point P1 and P2, and P1 and P3, respectively.

Now check the following inequality:

y12(0) < 0 < y13(0)

or

y12(0) > 0 > y13(0)

the result is yes if either of the above is true, if not check case 2.

or case 2:

x2 <= 0 <= x3

now use linear interpolation to obtain an equation y23(x), and y13(x). Where these are equations of line that pass through point P2 and P3, and P1 and P3, respectively.

Now check the following inequality:

y13(0) < 0 < y23(0)

or

y13(0) > 0 > y23(0)

the result is the origin is inside the triangle if either of the above is true

Let ABC be the triangle and P be any arbitrary point in 2D space.

The if P is inside the triangle ABC or not can be found by checking the signs of the following cross products:

AB X AP, BC X BP and CA X CP.

If the signs are same then it means the point is inside the triangle, o.w. it is outside the triangle.

For this question mark P as (0,0) and check for the signs of the above cross products.

Let ABC be the triangle and P be any arbitrary point in 2D space.

The if P is inside the triangle ABC or not can be found by checking the signs of the following cross products:

AB X AP, BC X BP and CA X CP.

If the signs are same then it means the point is inside the triangle, o.w. it is outside the triangle.

For this question mark P as (0,0) and check for the signs of the above cross products.

The area calculating method is clever, but mathemtically more intensive.

basically, assume the points are p1, p2 and p3, and we need to check if t is inside the tri-angle. We need to check if t is on the same side of line p1p2 with p3, same side of p2p3 with p1 and same side of p1p3 with p2.

1. let p1=(x1, y1), p2=(x2, y2), p3=(x3, y3) and t(xt, yt)

2. let k12=(y2-y1)/(x2-x1), k23=(y3-y2)/(x3-x2), k13=(y3-y1)/(x3-x1)

3. if (yt-k12*xt)/(y3-k12*x3) > 0, (yt-k23*xt)/(y1-k23*x1) > 0 and (yt-k13*xt)/(y2-k13*x2) >0, we can conclude t is inside the tri-angle

4. in case any line segment is vertical to x axile (for example x1=x2), and k is undefined, check (xt-x1)/(x3-x1) instead

Let the given points which form a triangle are A,B,C and let the point be P if

- khunima April 20, 2014Area(ABC) =Area(PAB)+Area(PBC)+Area(PAC) then the point P lie inside

the triangle , in this case P=(0,0).