Proving collinearity by constructing homotheties
by XmL, Jul 26, 2014, 4:55 AM
I got the following two problems from a really good friend of mine(a good geometer too), and I happen to solve them using "similar" methods(pun intended).
First Problem:
Problem Statement: Let
denote the reflection of
over
, let the line tangent to
at
intersect
at
, prove that
.

Solution
Second Problem:
Problem Statement:
is the circumcircle of
,
moves on segment
, the circle with
as diameter meets
at S,
, H is the orthocenter of
,
meets the line perpendicular to
through
at
. Prove that
is a fixed point.

Solution
First Problem:
Problem Statement: Let









Solution
Diagram: 
Proof: Reflect
over
to obtain
, hence
and now it suffices to prove that
lies on
. Let
, construct
on
resp. such that
are homothetic at
, Hence
and
are homothetic
. Since
, therefore
is tangent to
and we've proven
lies on 

Proof: Reflect



















Second Problem:
Problem Statement:














Solution
Diagram:
Proof: If
to moved to coincide with
, then we can see that
, a fixed point, would have to lie on the
-altitude. Hence we define
is the intersection of the line perpendicular to
through
with
-altitude, and we will prove that
are collinear.
Since
, this motivates us to construct points
and now it suffices to prove that
are homothetic
are concyclic .Now note that
, hence
are concyclic. Since
, therefore
are concyclic and we done

Proof: If









Since








This post has been edited 3 times. Last edited by XmL, Aug 12, 2014, 11:13 PM