Home > Shapes > Geometry
Parallelograms (Posted on 2017-09-09 )

Let ABC be an acute triangle with altitudes AD and BE. The
intersection of AD and BE is H. Points F and G make CADF
and CBEG into parallelograms. M is the midpoint of FG.
Ray MC intersects the circumcircle of ΔABC again at point N.
Prove that ANBH is a parallelogram.

Submitted by Bractals
No Rating
Solution:
(Hide )
The points M, C, and O (the circumcenter of ΔABC) are collinear
if the distance from M to the line OC is zero. This is true if
|CF|*sin(∠MCF) = |CG|*sin(∠MCG) (1)
From CADF and CBEG (1) becomes
|AD|*sin(∠MCF) = |BE|*sin(∠MCG) (2)
At point C we see that ∠MCF + ∠OCB = 90° and
∠MCG + ∠OCA = 90°. Thus,
sin(∠MCF) = cos(∠OCB) and sin(∠MCG) = cos(∠OCA)
Equation (2) then becomes
|AD|*cos(∠OCB) = |BE|*cos(∠OCA) (3)
From ΔADC and ΔBEC:
|AD| = |AC|*sin(∠ACD) and |BE| = |BC|*sin(∠BCE)
Since ∠ACD = ∠BCE, (3) becomes
|AC|*cos(∠OCB) = |BC|*cos(∠OCA)
This in turn becomes,
|AC|*(|BC|/(2*|OC|)) = |BC|*(|AC|/(2*|OC|)).
Therefore, Points M, C, and O are collinear.
Since CN contains point O, it is a diameter of the circumcircle.
Therefore, ∠CAN and ∠CBN are right angles since they are
subtended by CN.
AN and BE perpendicular to AC ⇒ AN || BE ⇒ AN || BH
BN and AD perpendicular to BC ⇒ NB || DA ⇒ NB || HA
Therefore, ANBH is a parallelogram.
QED

Comments: (
You must be logged in to post comments. )

Please log in:
Forums (0)
Newest Problems
Random Problem
FAQ |
About This Site
Site Statistics
New Comments (9)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On
Chatterbox: