# Projective geometry and planes equations

Let $$d_1, d_2$$ and $$d_3$$ be three non coplanar concurrent lines and $$O=d_1\cap d_2\cap d_3$$. In each line we place three points (2 by 2 distinct) $$A_i,B_i$$ and $$C_i$$, $$i\in \{1,2,3\}$$such that $$A_i\in d_1$$, $$B_i\in d_2$$ and $$C_i\in d_3$$. We suppose that $$\frac{2}{\overline{OA_2}}=\frac{1}{\overline{OA_1}}+\frac{1}{\overline{OA_3}}$$ and two analogous relations.

1. Find all planes $$A_iB_jC_k$$ so that $$i+j+k=6$$

2. Find equations of each plane by using an adequate coordinates systems.

3. We suppose that three of the planes found at 1. have a common point, show that all planes are concurrent.

For the first question I’ve found planes $$A_1B_2C_3$$ $$A_1B_3C_2$$. seven planes at all.

• None has a solution or a hint. – HAMIDINE SOUMARE Mar 4 at 23:45