# Thread: Finding length of sides of triangles using differences in distance to a common point

1. ## Finding length of sides of triangles using differences in distance to a common point

I have three co-linear points A, C, & B respectively. The distance between each of the points is known.

I have a point P at an unknown location, perhaps on the same line if possible, but usually always not co-linear with ACB.

The distance from P to C, called 'c', is unknown.

The distance from P to A is a + c, where 'a' is known (and 'c' is still the unknown distance from P to C). I.E., the distance from P to A is 'a' further than 'c' from P to C.

The distance from P to B is b + c, where 'b' is known (and 'c' is unknown). I.E., the same story as with point A.

a and b can be negative, but a+c and b+c are > 0 of course.

What is the distance from P to C ('c')?

Part II
Let's say the distance from P to A is unknown, but the difference to C and the difference to B are known. What is the distance to A?

Thanks,
Dale

2. using the law of cosines we can write 2 equations.

let the distance from AC=x and CB=y and these are known.

$E_1:$ $(a+c)^2=c^2+x^2-2cx\cos(\theta)$

$(b+c)^2=c^2+y^2-2cy\cos(\pi-\theta)$

using the subtaction Identity for cos gives

$(b+c)^2=c^2+y^2-2cy( \cos( \pi ) \cos( \theta )+\sin( \pi )\sin(\theta))$

this finally gives us

$E_2:$ $(b+c)^2=c^2+y^2+2cy\cos(\theta)$

$E_1$ and $E_2$ give a system of equations with the only unknowns c and theta...

I hope this helps

3. Thanks for the help but I think I need a little more help on this. I do not know the value of theta.

All I know is the distance between points A, C, & B and I know that from point P, A is 'a' further from P than C is from P and that B is 'b' further from P than C is from P.

Thanks,
Dale

4. Originally Posted by DSG
Thanks for the help but I think I need a little more help on this. I do not know the value of theta.

All I know is the distance between points A, C, & B and I know that from point P, A is 'a' further from P than C is from P and that B is 'b' further from P than C is from P.

Thanks,
Dale

You could isolate c in the first equation and sub it into the second equation and then solve for theta.

I didn't do the work by hand but maple gives the solution.

Good luck.

5. Thanks for your help. With the equation solved for c, I presume I do not need the equation for theta (since I do not care about the angle; I only need the distance from P to A, B, & C).

Thanks,
Dale

6. That's correct. You have 2 equations with 2 unknowns. Using substitution it is possible to eliminate 1 unknown to establish a formula for the other unknown; in this case the desired c.

7. The Empty Set and I Know One,