point inside an equilateral, that any line divides half areas (Chile 2016 L1 P2)

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point inside an equilateral, that any line divides half areas (Chile 2016 L1 P2)

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#1 h May 26, 2019, 8:42 PM • 1 Y

Y by Adventure10

For an equilateral triangle $\triangle ABC$ , determine whether or not there is a point $P$ inside $\triangle ABC$ so that any straight line that passes through $P$ divides the triangle $\triangle ABC$ in two polygonal lines of equal length.

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#2 h May 26, 2019, 9:10 PM • 2 Y

Y by Adventure10, Mango247

Answer?Click to reveal hidden text

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brainiacmaniac31

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brainiacmaniac31

#3 h May 26, 2019, 11:54 PM • 2 Y

Y by Adventure10, Mango247

How can you divide a triangle into 2 equal lines??

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ChickenAgent2227-_-

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#4 h May 27, 2019, 5:31 AM • 1 Y

Y by Adventure10

Basically, is there a point $P$ so that any straight line that passes through $P$ divides the triangle $\triangle ABC$ in two polygonal of equal perimeter?

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#5 h May 27, 2019, 6:04 AM • 1 Y

Y by Adventure10

I think the OP meant to find point $P$ inside $\triangle ABC$ so that any straight line passing though $P$ divides the perimeter of the triangle $\triangle ABC$ into two lines of equal length.

And Answer

This post has been edited 2 times. Last edited by Z_Math404, May 27, 2019, 6:44 AM

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#6 h May 27, 2019, 6:53 AM • 1 Y

Y by Adventure10

We consider three cases.

First-

When P is not on any median of the triangle. Then by simple making the line AP which intersects BC at Q, we can show that this is not the required point.

Second-

When P is on a median but not on the centroid. Let us say it is on the median from vertice A. Then draw a parallel line to BC through P and you can prove that it is not the point we are looking for.

Third-party

When P is on the centroid, do a Cartesian bash with a line that makes 45° with AB.

Result: No such point P exists.

This post has been edited 1 time. Last edited by Pi-is-3, May 27, 2019, 6:54 AM

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