# Equation of Plane through Center

I have a pentagon for which 3 vertices were chosen to compute the equation of the plane. How to I find the normal passing through its center?

$$P_1 = [ 3.096, \ 0.492, \ 3.287]$$

$$P_2 = [ 3.118, \ 0.227, \ 4.669]$$

$$P_3 = [ 2.214, \ 1.476, \ 2.801]$$

Equation of Plane: $$2.809x + 0.732y - 3.586z = 4.614$$

Given this equation how do I find the normal passing through the center of the pentagon?

• Do we know what kind of vertices are $P_1,P_2,P_3$. Are the consecutive or random? – gimusi Sep 5 '18 at 14:47
• @gimusi P1 is the first vertex, P2 is the second in the clockwise direction and P3 is the last (so the first vertex counter clockwise from P1). – EA00 Sep 5 '18 at 14:50
• the key point is to find $P_4$ and $P_5$ – gimusi Sep 5 '18 at 14:59

• from the equation of the plane $ax+by+cz=d$ we know that $n=(a,b,c)$
• find $P_4$ and $P_5$ by vectors
• the center of the pentagon is given by $C=\frac{P_1+P_2+P_3+P_4+P_5}5$