Task. Given three line segments $a$ , $b$ and $c$ . Using compass and straightedge, construct the fourth proportional of the line segments.

Solution. Draw an angle ($\alpha$ ) and denote its vertex by $P$ . Separate from one side of the angle the line segments $PA = a$ and $AB = b$ , and from the other side of the angle the line segment $PC = c$ . Draw the line $AC$ and another line parallel to it passing through $B$ . If the last line intersects the other side of the angle in the point $D$ , then the line segment $CD = x$ is the required fourth proportional: $$a:b \;=\; c:x$$ Justification: the intercept theorem.

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