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I need an Excel formula that can help me find the third point in a triangle, given most other information. This is not homework, meaning I do not need to do this just once. I need a consistent way to accomplish it repeatedly over the next few months, as quickly and accurately as possible, so I am hoping Excel or something similar will be the best tool. I am under the impression that there may always be two valid answers to this problem. If so, and if possible, four formulas, two for the possible X values, and two for the possible Y values, would be fine. In most cases, working out which is correct will not be difficult. My main concern is being able to do this accurately, repeatedly.

All points are on a 2D Cartesian system, (have X and Y values), and can be negative or positive. The triangle will be random, and so must be assumed to be scalene. For each triangle, I have two Cartesian points, along with the distances from those points to the third point. From this I can easily ensure that in the other columns, I have the lengths of all three sides, and all three inner angles in both radian and degree formats, as well as the two Cartesian points as X and Y values. How can I use an Excel formula to come up with the remaining X and Y values of the two valid third points?

I’ll lay out the columns / variables that I have here to be clear, in case you’d like to use my letters rather than traditional notation.

point 1:
A = X1
B = Y1

point 2:
C = X2
D = Y2

E = distance from p1 to p2  (calculated from other values)
F = distance from p1 to p3
G = distance from p2 to p3

H = inner angle at p1 in radians (calculated from other values)
I = inner angle at p2 in radians (calculated from other values)
J = inner angle at p3 in radians (calculated from other values)

The cells that will contain my formulas:
K = possible X3
L = other possible X3
M = possible Y3
N = other possible Y3

My assumption is that there would be two formulas for X3, and two formulas for Y3, each producing a possible solution, but I’m not 100% sure if this is true.

I can’t quite handle the math, but my instinct would be that I could somehow grab one of my distance formulas, for example the one to calculate distance E:


and manipulate it to solve for the one X,Y set, given distance and the other X,Y set, but each time I attempt that, I get lost or make an error before I find a functioning formula.

Is this possible to do in a simple way using the columns / variables above and a set of four Excel formulas? Thank you.

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Not sure, but it sounds like Wolfram Alpha might be useful for this. – David Jun 14 '13 at 21:00

3 Answers 3

up vote 5 down vote accepted

Here is one possible set of formulas, that does basically what Dane described. Tested in Google Docs.

A= ____                         // X1
B= ____                         // Y1
C= ____                         // X2
D= ____                         // Y2
E= sqrt((A-C)^2+(B-D)^2)        // Distance 12
F= ____                         // Distance 13
G= ____                         // Distance 23
H= acos((E^2+F^2-G^2)/(2*E*F))  // Angle 1
I= acos((E^2+G^2-F^2)/(2*E*G))  // Angle 2
J= acos((F^2+G^2-E^2)/(2*F*G))  // Angle 3
K= A+F*cos(atan2(C-A,D-B)+H)    // X3
L= A+F*cos(atan2(C-A,D-B)-H)    // X3Alt
M= B+F*sin(atan2(C-A,D-B)+H)    // Y3
N= B+F*sin(atan2(C-A,D-B)-H)    // Y3Alt
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Looks good. Except you switched columns L and M from what Gusy asked for (but I believe that your layout makes more sense; it’s consistent with the arrangement of columns B and C). – Scott Jun 15 '13 at 0:39
Yep. And that worked perfectly for me. I just finished testing a large set and that was a huge time-saver. Thank you! – Gusy Jun 17 '13 at 18:43
@scott: Yeah, it did not even occur to me to check that. I just Assmumed it was X3, Y3, X3Alt, Y3Alt. I'll update my answer just for the sake of minimizing confusion. – Kevin Cathcart Jun 17 '13 at 18:45

You could just plot the circles with radius given from each known point and then see where they intersect. Like shown below.

enter image description here

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To calculate your third point, you need the following:

  1. A starting point
  2. The distance
  3. The angle from the x axis

You will then use Excel's =sin() or =cos() functions and the math explained in this Mathematics question.

The trick is calculating (3) the angle from the x axis. This Stack Exchange question tells you how to determine the angle from the x axis on your known side. You will use Excel's =atan2(). From this, you will either add or subtract your interior angle (your value H, for example). Adding vs. subtracting here is what will give you your two different points.

Be careful about using Radians vs. Degrees.

Have fun and let us know what you come up with!

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