# Find a point on a plane

I have three points to write the equation of a plane: assume $P_1=(x_1,y_1,z_1),P_2=(x_2,y_2,z_2),P_3=(x_3,y_3,z_3)$. I can also write the equation of this plane.

I want to obtain the coordinate of a point $(P_4)$ in this plane which is at a particular distance of $P_1, P_2$, whereas $P_1,P_2,P_4$ constitute a right angled triangle with each other.

Could anyone help me with this?

• Do you know that if $P$ and $Q$ are at opposite ends of the diameter of a circle, and $R$ is any other point on that circle, then the triangle $PQR$ has a right angle at $R$? – Gerry Myerson Sep 23 '13 at 13:00
• If you know that $P_4$ is in that plane and you have the distance between $P_4$ and $P_1$, the distance between $P_4$ and $P_2$, then there are (at most) two possibilities for $P_4$ and the triangle $P_1, P_2, P_4$ may not be right-angled. – njguliyev Sep 23 '13 at 13:13

As Gerry Myerson mentioned, for any point $P_4$ that lies on a circle where $\overline{P_1P_2}$ forms the diameter of the circle, the triangle $P_1P_2P_4$ will have a right angle at $P_4$. However, if you know the distances between $P_4$ and $P_1,P_2$ each, you will find two solutions which may or may not form a right angled triangle.. Do you have the absolute distances, or maybe a ratio of distances?

By the way, another way to get right angles would be at $P_1$ or $P_2$.

EDIT:

According to your comment, you have the following conditions:

1. $(P_4-P_2)^2 = d^2$, so $P_4$ has to be in a sphere with radius $d$ around $P_2$, or on a circle on the plane you already found

2. To get a right angle in $P_2$, you can choose $P_4$ to be on a line through $P_2$ perpendicular to $\overline{P_1P_2}$ ($P_4$ would be on the intersection of the line and the circle/sphere, so there are two solutions). There are more possible solutions (you could get a line through $P_1$ to get a right angle in $P_1$, but you will have no guarantee that the sphere and this line have an intersection, or you could try to find an intersection of the sphere with the circle where $\overline{P_1P_2}$ is the diameter, but also here there's no guarantee you'll find a solution).

Hope this helps!

• I have only the absolute distance of P2 and P4. In this way, do we have only one answer or more? – Maryam H Sep 23 '13 at 13:45
• Would you please help me with obtaining the coordinate of point P4? I want to write a Matlab code that according to the point of P1,P2, P3 give me the coordinate of P4. – Maryam H Sep 23 '13 at 13:48
• I edited my answer according to the new information you gave. You might want to edit your question to make clear which distance you know (from $P_2$ to $P_4$). – Lisa Sep 23 '13 at 14:07