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# construction of fourth proportional

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.

The below picture illustrates this solution:

Note. The special case $c=b$ gives the third proportional $x$ of $a$ and $b$:

$a:b\;=\;b:x$ |

## Mathematics Subject Classification

51M15*no label found*51M04

*no label found*

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