Nice Geometry Problem!

ZHero · 2010-05-28 23:51:00

Hi all...
This is a nice Geometry Problem proposed by one of my close friends!
I'm really stuck and need some help....
Thanks!

gurthbruins · 2010-05-30 00:25:18

Let P, Q be the points where the tangent touches the bigger and smaller arcs.
Let AM intersect XY at Z.
Compute length AM.
AZ = 2(MZ)...... (ratio of the 2 similar rt-angled triangles); so AZ = 2/3 AM.

Solve the triangles APZ and MQZ.
Compute angles BAZ and AZX: gives the slope of XY.

We know XY goes through the point P which we can compute.
So we can get the equation of line XY.

Solve against the equations for FE and CD to get the coordinates of X and Y.
From these compute length XY.

ZHero · 2010-05-30 19:59:52

I showed him your reply and...
mmmm... Can we try solving it using Euclidean Geometry only?
Sort of... NO USE of co-ordinate system and vectors!
(he says he does 'em using Similar Triangles and Trigonometry )

Thanks for looking into it!

ZHero · 2010-05-30 23:26:49

Here is another (and perhaps Tougher) version of the same question...
Try calculating the answer for this and see!!

ZHero · 2010-05-30 23:31:21

He is looking for a General Solution for the lengths of all such XY's for all Regular Polygons (no. of sides > 4)!
I don't know if thats possible or not...
May be, he'll come up with a General Approach instead!?

Fruityloop · 2012-05-23 19:34:24

I used the idea from gurthbruins to solve this problem.
This is a very hard problem. I had printed out the problem on a sheet of
paper and decided to try to solve it. First, we draw a straight line connecting
points A and M. The point where this line intersects XY we'll call point Z.
We'll make another point called G which is the mid-point of AB. The point of
tangency for the big circle we'll call W. The center of the hexagon is the origin.
Notice that

We need to know what the coordinates of Z are. The x-coordinate is

so the coordinates of point Z are

So we must solve the linear equation

simultaneously with the equations
in order to get the
coordinates of X and Y. Many, many steps later one ends of with
the x and y coordinates of ponts X and Y as follows...

using to get the distance of XY we finally end
up with after many more steps.. drum roll please.....
as distance of XY.
Then afterwards be sure to take some Advil for the headache and
a tissue to wipe the blood now oozing from your eyes.

Last edited by Fruityloop (2012-05-23 20:50:23)

Math Is Fun Forum

#1 2010-05-28 23:51:00

Nice Geometry Problem!

#2 2010-05-30 00:25:18

Re: Nice Geometry Problem!

#3 2010-05-30 19:59:52

Re: Nice Geometry Problem!

#4 2010-05-30 23:26:49

Re: Nice Geometry Problem!

#5 2010-05-30 23:31:21

Re: Nice Geometry Problem!

#6 2012-05-23 19:34:24

Re: Nice Geometry Problem!

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