#8) ELMO Coordinate Bash
by TheDarkPrince, Feb 11, 2018, 6:03 PM
Problem: Let 
be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that 
and 
are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
Proposed by Michael Ren
be a triangle with orthocenter 
and let 
be the midpoint of 
Suppose that 
and 
are distinct points on the circle with diameter 
different from 
such that lies on line 
Prove that the orthocenter of 
lies on the circumcircle of 
Proposed by Michael Ren
be the orthocenter of 
lie on the circle with diameter 
,
.
and 
.
and 
.
.
is a parallelogram.
.
.
:
.
,
.
.
is parallel to the 
,
.
.
be the center of circle with diameter 
.
.
.
be 
.
Simplification of coordinates of 
.
,
we get after simplifying, 
Coordinates of 
.
we get, 
Equation of circumcircle of 
be the center of circle 
.
,
.
.
be the midpoint of 
.
.
,
.
.
.
.
.
Putting the coordinates of 
We cancel the denominators.
Simplifying and taking similar terms together we get,
So, This post has been edited 2 times. Last edited by pro_4_ever, Mar 1, 2018, 10:46 AM
March 2018
February 2018