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Barukh
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 Reconstruct Triangle from Points (Geometry)   « on: Sep 24th, 2007, 12:46am » Quote Modify

A well-known class of geometric construction problems requires construction of a triangle from three given parts (sides, angles, cevians, etc.).  I believe several problems from this class were presented at this forum.

The less known but not less fascinating is construction of a triangle from three given points. Here’s one:

Given 3 points that are feet of the altitudes of a triangle, reconstruct the triangle.
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Eigenray
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #1 on: Sep 24th, 2007, 3:49am » Quote Modify

Three weeks ago, I wouldn't have known how to do this
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Barukh
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #2 on: Sep 24th, 2007, 4:20am » Quote Modify

on Sep 24th, 2007, 3:49am, Eigenray wrote:
 Three weeks ago, I wouldn't have known how to do this

I am glad you learnt something new with a little help from me...

BTW, how many solutions are there?
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Eigenray
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #3 on: Sep 24th, 2007, 7:41am » Quote Modify

Looks like 4, but could also be 1 or infinity in degenerate cases.
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Barukh
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #4 on: Sep 24th, 2007, 9:36am » Quote Modify

on Sep 24th, 2007, 7:41am, Eigenray wrote:
 1 or infinity in degenerate cases.

What are the cases when you get a single solution?
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Eigenray
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #5 on: Sep 24th, 2007, 10:52am » Quote Modify

on Sep 24th, 2007, 9:36am, Barukh wrote:
 What are the cases when you get a single solution?

When the points are collinear?
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Barukh
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #6 on: Sep 26th, 2007, 1:15am » Quote Modify

on Sep 24th, 2007, 10:52am, Eigenray wrote:
 When the points are collinear?

Right.

Here is another one:

2. Given: one vertex, incenter, circumcenter.
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Eigenray
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #7 on: Sep 26th, 2007, 3:39am » Quote Modify

Here is a rather uninspired solution:

 hidden: Change coordinates so that the vertex is at O=(0,0), the incenter I=(1,0), and the circumcenter C=(a,b).  Construct lines through O of slope m.  For each line, mark the point Xi on the line such that the perpendicular bisector of OX goes through C.  Then we require that angle OX1I = angle IX1X2.  Solving for Xi in terms of m, and equating the tangents of the angles, gives m = (2a-1)/{4b2+4a-1}, which of course is constructible.

But I'm quite sure that's not what you had in mind.
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Barukh
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #8 on: Sep 26th, 2007, 3:51am » Quote Modify

on Sep 26th, 2007, 3:39am, Eigenray wrote:
 But I'm quite sure that's not what you had in mind.

Confirmed.
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Aryabhatta
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #9 on: Sep 27th, 2007, 12:59am » Quote Modify

on Sep 26th, 2007, 3:51am, Barukh wrote:
 Confirmed.

Another method:

Use the fact that the distance between the incenter and circumcenter is sqrt(R(R-2r)) where R is the circum-radius and r is the in-radius.

Using this we can draw the incircle and circumcircle. The tangents to the incircle through one vertex and their intersections with the circumcircle determine the triangle.
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Barukh
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #10 on: Sep 27th, 2007, 6:19am » Quote Modify

Yes, exactly. Good job, Aryabhatta!

BTW, how would you constuct the incircle?
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Aryabhatta
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 Re: Reconstruct Triangle from Points (Geometry)   « Reply #11 on: Sep 27th, 2007, 11:09am » Quote Modify

on Sep 27th, 2007, 6:19am, Barukh wrote:
 Yes, exactly. Good job, Aryabhatta!     BTW, how would you constuct the incircle?

To construct the circle, we construct the radius first. To do that we construct x = R-2r. Let y = R

Then we are given sqrt(xy) and y. We use the well known fact that the perpendicular on the hypotenuse of a right triangle is of length sqrt(xy) where x and y are the lengths of the two segments of the hypotenuse so formed because of the perpendicular on it.

Given sqrt(xy) and y, we can easily construct the whole triangle and, as a result, we can construct x.
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