How to calculate the third vertex from two coordinates of a triangle

Updated April 17, 2017

Any three points on a plane define a unique triangle. From two known points, an infinite number of triangles can be formed simply by arbitrarily choosing one of the infinitely many points on the plane as the third vertex. Finding the third vertex of a right triangle, an isosceles triangle, or an equilateral triangle, however, takes a little more calculation.

Divide the difference between your two points' "y" coordinates by the difference between their respective "x" coordinates. This gives the slope of the line between your two points, or "m." For example, if your points are (3,4) and (5,0), the slope is 4/-2, so m = -2.

Multiply "m" by the "x" coordinate for one of your points, and then subtract that from the "y" coordinate of the same point to get "a." The formula for the line connecting your two points is y = mx + a. In the example above, y = -2x + 10.

Find the formula of the line perpendicular to the line between your two known points, which runs through each of them. The slope of a perpendicular line is equal to -1/m. You can find the value for "a" by substituting the "x" and "y" from the appropriate point. For example, the perpendicular line running through the first example point above would have the formula y = 1/2x + 2.5. Any point on one of these two lines will form the third vertex of a right triangle with the other two points.

Find the distance between your two points using Pythagoras' theorem. Take the difference between the "x" coordinates and square it. Square the difference between the "y" coordinates, and add the two squares together. Then take the square root of the result. It is the distance between your two points. In the example, 2 x 2 = 4, and 4 x 4 = 16, the distance is equal to the square root of 20.

Find the midpoint between your two points, which has the coordinates halfway between the coordinates of the known points. In the example, this is (4,2), because (3+5)/2 = 4 and (4+0)/2 = 2.

Find the formula for a circle centred on the midpoint. The formula for a circle is in the form

(x - a)^2 + (y - b)^2 = r^2, where "r" is the radius of the circle and (a,b) is the centerpoint. In the example, "r" is half the square root of 20, so the circle formula is (x-4)^2 + (y-2)^2 = (sqr(20)/2)^2 = 20/4 = 5. Any point on this circle is the third vertex of a right triangle with the known two points.

Find the formula for the perpendicular line passing through the midpoint of the two known points. This will be y=-1/mx + b, and the value of "b" is determined by substituting the midpoint coordinates into the formula. In the example, the result is y = -1/2x + 4. Any point on this line will be the third vertex of an isosceles triangle with the two known points as its base.

Find the formula for a circle centred on either of the two known points with a radius equal to the distance between them. Any point on this circle forms the third vertex of an isosceles triangle, where the base is the line between this point and the other known circle -- the one which isn't the centre of the circle. Also, where this circle intersects the midpoint perpendicular is the third vertex of an equilateral triangle.

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About the Author

Tom Kantain has been writing and editing in various forms for over 20 years. He has written a regular magazine column on the philosophy of games. Kantain holds a Master of Arts in philosophy as well as a Bachelor of Laws.