Find point P on line BC equidistant to points A and O without calculation.

Question

Consider the following points $A(5,3,3)$ $ B (2,6,2)$ $ C(3,8,2)$ $O(0,0,0)$. The problem asks to construct a point P on line BC that is equidistant to A and O using projections on a frontal and horizontal plane. The problem should be easy to solve by calculation by taking a point P on line BC and then solving the equation PA=PO, but i have no idea how to solve this problem using no calculation.

Thanks for the help!

Answer 1

Here is a solution where you use only projections on plane xy and plane OAA' - the plane orthogonal to xy.

See the image, all the projections on xy plane are drawn red, and projections on OAA' are black

You start from the red A' and end on the P"'.

A' - projection of A on xy

S - mid point of OA

P' - projection of P on OAA'

P" - projection of P' on xy

P"' - projection of P on xy