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Say I have a triangle with points A, B, and C, and I know the lengths of AC, AB, and BC. The triangle may or may not be a right triangle.

Example: click here, I can't embed the image since I don't have 10 reputation yet

I decide to label BC as the X axis, and a line extending through the altitude of A as the Y axis (origin is where the red line intersects with BC)

How can I find the coordinates of a point D (ideally inside OR outside the triangle if possible) with just side lengths of the triangle and distances AD, BD, and CD?

I understand basic trigonometry and basic hyperbolic equations, but can't find a solution. An explanation would be awesome. Thank you!

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This seems pretty lengthy, but you could split the triangle up into the following right triangles (see image). Note that the dashed red lines are perpendicular to the sides of the triangle.

enter image description here

Then with Pythagoras' theorem and basis trigonometry you can determine all the lengths that you might need.


Alternatively, if $B$ is the origin instead, then the distance $BD$ is the length of the vector from the origin to $D$, and you can use Pythagoras to determine the precise vector after using the angle formulae. If you wanted to keep $A$ as crossing the $y$-axis, then shifting that solution left by cosine of the angle $ABC$ gives you the coordinates. Note that in both cases, the height of $D$ is always the length of $z$.

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  • $\begingroup$ Thanks! Is there any way to get the height of the altitudes with a simple formula? $\endgroup$
    – Ben Gubler
    May 3 '18 at 8:04
  • $\begingroup$ Not quite, only the height of $A$ is quick to determine by using the cosine formula to get angle $ABC$, and then the height of $A$ is $AB$ times the sine of that angle. The rest will be fairly lengthy to determine by using your basic trigonometry identities on each of the smaller six triangles and solving them simultaneously. $\endgroup$ May 3 '18 at 8:52

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