Track / Stage 5 / 336 of 400 #936 of 1964

Problem 936

AIME late

Geometry Difficulty 5.9 Prove it

1.14. Points $A_{1}$ and $A_{2}$ belong to planes $\Pi_{1}$ and $1_{2}$ , intersecting along line $l$ . Prove that the line $A_{1} A_{2}$ forms equal angles with planes $\Pi_{1}$ and $\Pi_{2}$ if and only if points $A_{1}$ and $A_{2}$ are equidistant from line $l$ .

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

1.14. Consider the projection onto the plane П, perpendicular to the line $l$ . Points $A_{1}$ and $A_{2}$ are projected to $A_{1}^{\prime}$ and $A_{2}^{\prime}$ , the line $l$ is projected to the point $L$ , and the planes $\Pi_{1}$ and $\Pi_{2}$ are projected to the lines $p_{1}$ and $p_{2}$ . As follows from the solution to problem 1.11, the line $A_{1} A_{2}$ forms equal angles with the perpendiculars to the planes $\Pi_{1}$ and $\Pi_{2}$ if and only if the line $A_{1}^{\prime} A_{2}^{\prime}$ forms equal angles with the perpendiculars to the lines $p_{1}$ and $p_{2}$ , i.e., forms equal angles with the lines $p_{1}$ and $p_{2}$ themselves; and this, in turn, means that $A_{1}^{\prime} L = A_{2}^{\prime} L$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.