269. Some nice and easy properties in a trapezoid.
by Virgil Nicula, Apr 25, 2011, 12:41 AM
PP1 (O.M. - Moscova, 2011). Let 
be a trapezoid with 
. Denote the midpoints 
and 
of
![$[AB]$](//latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and ![$[CD]$](//latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
respectively. Define the point 
for which 
, 
. Prove that 
.
Proof. Let midpoint
of ![$[AD]$](//latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
. Observe that is orthocenter of 
. In conclusion, 
, i.e. 
.
PP2. Let a trapezoid , where 
. Denote 
and the midpoint of
side . The circumcircles of 
, 
intersect at 
. Prove that 
.
Proof.
.
Denote 
. Apply Sinus' theorem in 
, i.e. 
.
In 
apply an well-known property 
. From 
, 
, 
obtain 
is symmedian in .
Remark. If denote
, then the ray 
is symmedian in the triangle 
.
PP3. is a trapezoid with 
. Let 
and 
so that
and ![$[APQB]=[CDPQ]$](//latex.artofproblemsolving.com/d/a/a/daa4aa004b5930b433c727d3630d2876a28123a7.png)
. Prove that 
.
Proof. Let
. Observe that 
and ![$[APQB]=[CDPQ]\iff$](//latex.artofproblemsolving.com/5/b/f/5bf0bdc244f79b0be8e59ae2f96b7a66c04c8fbc.png)
![$\frac {[ABCD]}{[CDPQ]}=2\iff$](//latex.artofproblemsolving.com/0/8/d/08de024318c2f149968d834a4423b851ee8c0904.png)
.
Remark.
, where 
. Generally, ![$[CDPQ]=\rho\cdot [ABQP]\ \iff\ \rho\cdot a^2+b^2=(\rho +1)\cdot r^2$](//latex.artofproblemsolving.com/8/4/c/84c8bfbb761dd2a5d2900ee306588838a00c9a19.png)
.
be a trapezoid with 
. Denote the midpoints 
and 
of![$[AB]$](http://latex.artofproblemsolving.com/a/d/a/ada6f54288b7a2cdd299eba0055f8c8d19916b4b.png)
and ![$[CD]$](http://latex.artofproblemsolving.com/e/7/0/e70960e9e5738a46ad23f794e796ef3cb4ad7e2c.png)
respectively. Define the point 
for which 
, 
. Prove that 
.Proof. Let midpoint
of ![$[AD]$](http://latex.artofproblemsolving.com/0/f/3/0f3e4c424371b27673db323ced8ef0777940c0d4.png)
. Observe that is orthocenter of 
. In conclusion, 
, i.e. 
.PP2. Let a trapezoid
, where 
. Denote 
and the midpoint ofside
. The circumcircles of 
, 
intersect at 
. Prove that 
.Proof.
.
Denote 
. Apply Sinus' theorem in 
, i.e. 
. In 
apply an well-known property 
. From 
, 
, 
obtain 
is symmedian in .Remark. If denote
, then the ray 
is symmedian in the triangle 
.PP3.
is a trapezoid with 
. Let 
and 
so that
and ![$[APQB]=[CDPQ]$](http://latex.artofproblemsolving.com/d/a/a/daa4aa004b5930b433c727d3630d2876a28123a7.png)
. Prove that 
.Proof. Let
. Observe that 
and ![$[APQB]=[CDPQ]\iff$](http://latex.artofproblemsolving.com/5/b/f/5bf0bdc244f79b0be8e59ae2f96b7a66c04c8fbc.png)
![$\frac {[ABCD]}{[CDPQ]}=2\iff$](http://latex.artofproblemsolving.com/0/8/d/08de024318c2f149968d834a4423b851ee8c0904.png)
.Remark.
, where 
. Generally, ![$[CDPQ]=\rho\cdot [ABQP]\ \iff\ \rho\cdot a^2+b^2=(\rho +1)\cdot r^2$](http://latex.artofproblemsolving.com/8/4/c/84c8bfbb761dd2a5d2900ee306588838a00c9a19.png)
.This post has been edited 15 times. Last edited by Virgil Nicula, Nov 22, 2015, 8:33 AM
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