unusual construction of angle bisector - 2016 Rusanovsky Lyceum Olympiad 8.4

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unusual construction of angle bisector - 2016 Rusanovsky Lyceum Olympiad 8.4

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#1 h Jul 5, 2021, 9:14 AM

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Let's call a "mermaid" a device that builds the geometric location of points equidistant from the point and this line. Construct the bisector of the given angle $ABC$ with an inaccessible vertex using a "mermaid" and a ruler.

(T. Timoshkevich)

This post has been edited 2 times. Last edited by parmenides51, Jul 5, 2021, 9:20 AM

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#2 h Jul 6, 2021, 12:37 PM

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A "mermaid" juste draws parabolas !

On the picture, only the inside of the greyed rectangle is accessible (simulating a sheet of paper)
all that is outside (dotted lines) is "inaccessible" and can't be used for any kind of construction
Let $P$ any accessible point "between" lines $(AB)$ and $(AC)$ and "mermaid" this point with the two [ part of ] sides $AB$ and $AC$
The two mermaid loci [ arc of parabolas ] intersect at $I$ and $J$
If one of these points is outside (inaccessible) just repeat with another $P$

$PJ = JR$ an $PJ = JT$ hence $JR = JT$ and $J$ is then a point on the searched angle bisector of $\angle A$
same for $I$ or any such constructed point from another $P'$
hence the bisector is line $(IJ)$
( $R$ and $T$ are not to be constructed, just for the proof)
Choosing $P$ outside the region between the lines gives the external angle bisector. edit : fully inaccessible

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This post has been edited 1 time. Last edited by mathafou, Jul 6, 2021, 12:39 PM

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