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#26 Re: This is Cool » Theorem in Euclidian geometry » 2013-04-02 06:47:10
You're right, Bob. Thanks!
#27 Re: This is Cool » Theorem in Euclidian geometry » 2013-04-02 06:06:50
Thanks Bob;
one of the unpleasant things of mathematics is that sometimes someone is a step ahead of you.
I will try to demonstrate something else. At least mine is an alternative demonstration 
#28 This is Cool » Theorem in Euclidian geometry » 2013-04-02 01:01:27
- berliner
- Replies: 23
Hi everyone! I want to submit you a theorem in geometry.
Given a line and a point not on the line and two oblique carried out from the same point, distances from the midpoints of the oblique to the line are congruent.
Hypothesis: P doesnt belong to r wile H, A, R belong to r and PH is perpendicular to r. M is the midpoint of PA and N is the midpoint of PR.
Thesis: MS is congruent to NT
Begin to build the axis of symmetry t passing through the midpoint of MS.
St: S<->M.
r <-> r' and M belongs to r'
St: A <-> x. x is a point which belongs to r' because T and x correspond in an isometry which by definition maintains unchanged the distance of the points.
We must demonstrate that x is the midpoint of PR.
The symmetry transforms PR in P'R'. The midpoint of P'R' is T, because St: R<->R', P <-> P', PR<->P'R', PR = P'R' and NT is perpendicular to r at point T. So, since that T is the midpoint of P'R' and N is the midpoint of PR and since that PR and P'R' correspond then T and N correspond. So x = N and since x belongs to r' then also N belongs to r'. So M and N belong to r', S and A belong to r, r and r' are parallel (because parallelism is am invariant of symmetry) and MS and NT are perpendicular to r the MS and NT are congruent (because the distance between parallel lines doesn't change).