Complicated dependencies
by math_explorer, Dec 3, 2010, 10:41 AM
Problem:
In triangle
, 
is a median and 
is a cevian, and they intersect at 
. Prove that is the angle bisector of 
iff 
.
Solution:
Suppose the third cevian passing through is 
. By Van Obel's Theorem, 
. But 
by Ceva's Theorem, and obviously 
. The conclusion follows from the angle bisector theorem: bisects iff 
.
Problem:
is an inscribed quadrilateral. The diagonals intersect at 
. 
are the midpoints of 
and 
respectively; 
are the feet of the altitudes from to and 
, respectively. Prove that 
.
Obvious fact:
The incircle-side tangency points and the excircle-side tangency points are symmetric about the midpoint.
Obvious fact:
has only one real solution.
In triangle
, 
is a median and 
is a cevian, and they intersect at 
. Prove that is the angle bisector of 
iff 
.Solution:
Suppose the third cevian passing through
is 
. By Van Obel's Theorem, 
. But 
by Ceva's Theorem, and obviously 
. The conclusion follows from the angle bisector theorem: bisects iff 
.Problem:
is an inscribed quadrilateral. The diagonals intersect at 
. 
are the midpoints of 
and 
respectively; 
are the feet of the altitudes from to and 
, respectively. Prove that 
.Obvious fact:
The incircle-side tangency points and the excircle-side tangency points are symmetric about the midpoint.
Obvious fact:
has only one real solution.
May 2020
January 2020