[makar]

Let be an acute-angled triangle and let be the feet of perpendiculars from respectively to Let the perpendiculars from to meet them in respectively. Prove that the points are collinear.

[Dimitris X]

makar wrote:

Let be an acute-angled triangle and let be the feet of perpendiculars from respectively to Let the perpendiculars from to meet them in respectively. Prove that the points are collinear.

Nice.
Let be the orthocenter of the triangle.
Consider the complete quadrilateral .Then the Miquel point of the complete quadrilateral is the .So are collinear(Simson line ).

Note that is the Miquel point of the complete quadrilateral because:
1) is inscribed,so the miquel point is on the line and
2) is inscribed so the miquel point is on the line .

So the miquel point is the intersection of these two lines.

[makar]

http://www.artofproblemsolving.com/viewtopic.php?t=236802

[Moderator edit: also http://www.artofproblemsolving.com/viewtopic.php?t=291532 ]

[jayme]

Dear Mathlinkers,
the line PQMN is also named "the Miquel's line of..."
Sincerely
Jean-Louis

[makar]

jayme wrote:

Dear Mathlinkers,
the line PQMN is also named "the Miquel's line of..."
Sincerely
Jean-Louis

Dear friend,
Can you provide me some reference for the Miquel's line or some link which explain it in detail. It is totally new for me,though I already gone through Miquel's Point and Miquel's Theorem but never heard about Miquel's line.

thanks in advance,

regards

Makar

[Dimitris X]

makar wrote:

jayme wrote:

Dear Mathlinkers,
the line PQMN is also named "the Miquel's line of..."
Sincerely
Jean-Louis

Dear friend,
Can you provide me some reference for the Miquel's line or some link which explain it in detail. It is totally new for me,though I already gone through Miquel's Point and Miquel's Theorem but never heard about Miquel's line.

thanks in advance,

regards

Makar

Actually the ''miquel'' line is the simson line of the complete quadrialateral,which is parallel to Aubers line and perpendicular to Gaus's line