Question

Prove the Postulate of Pasch

Answer 1

Pasch's Postulate states that if a line intersects a triangle not at a vertex, then the line intersects two sides of the triangle.

i.e given a $\begin {aligned} \triangle \end {aligned}$ ABC and a line segment $\begin {aligned} l \end {aligned}$, if $\begin {aligned} l \end {aligned}$ intersects the segment AB, it also intersects the segment AC or BC.

\underline\bold{Proof}

Note:

1. $l$ divides plane $ABC$ into two distinct planes
2. Call the two planes and $H$1 and $H$₂
3. Point A and Point B are in different half planes since $l$ divides line AB
4. Let A $\varepsilon$ plane $H$₁ and B $\varepsilon$ plane $H$₂
5. C $\varepsilon$ $H$₁ or C $\varepsilon$ $H$₂

Case 1

Assuming Point C $\varepsilon$ $H$₁

Since Points A and C $\varepsilon$ $H$_1, $\therefore$ A and C are on the same side of the plane

This also means that $l$ does not intercept AC

Since C $\varepsilon$ $H$₁ and B $\varepsilon$ $H$_2, $\therefore$ B and C are on opposite sides of $l$

$\therefore$ $l$ intercepts BC

Case 2

Assuming Point C $\varepsilon$ $H$₂

Since Points B and C $\varepsilon$ $H$_2, $\therefore$ B and C are on the same side of the plane

This also means that $l$ does not intercept BC

Since C $\varepsilon$ $H$₂ and A $\varepsilon$ $H$_1, $\therefore$ A and C are on opposite sides of $l$

$\therefore$ $l$ intercepts AC

Therefore $l$ intercepts either AC or BC