Fontene Theorems Elementary Proofs
by XmL, Aug 12, 2014, 10:26 PM
In this blog we will prove all three of the Fontene theorems in an elementary fashion with a different point of view. Either you want to know what they are beforehand or try proving them yourself they are here: http://mathworld.wolfram.com/FonteneTheorems.html
Let's begin by defining an important point:
Let
be an arbitrary line that passes through the circumcenter of
, define
. The Simson lines of
wrt
meet at
.
Result: If
and
. Then the pedal circle of
wrt
passes through
.
Before reading on
Diagram
Proof: Let the pedal triangle of
be
, The
Simson lines
,
are similarly defined.
Since
is the diameter, we know that the Simson lines are orthogonal. From this we can discover a lot of similar triangles in the diagram(that can be proven by simple angle chasings). In particular,
. Note that since
, therefore
, hence
are corresponding points in the two similar triangles
, symmetrically we have
. Hence
are concyclic which means
and we are done.
.
Correlary-2nd Fontene theorem:
If a point moves on a fixed line through the circumcenter, then its pedal circle passes through a fixed point on the nine-point circle.
Proof: Clearly
defined above is the fixed point, moreover since the nine-point circle is the pedal circle of the circumcenter therefore
lies on it.
.
1st Fontene theorem(Rephrased): The
midline,
concur.
Proof: Let
be the midpoints of
,
. Since
, therefore
is the midpoint of
, likewise
is the midpoint of
. Now construct
, which is the miquel point of complete quad.
and the reflection of
over
. Hence we have

Next we will prove that
: It's easy to show that
. Let
. Since
are two corresponding points of the similar triangles mentions, therefore
and hence
.
Note that
and that
are two corresponding points, therefore
, since
, therefore
are collinear.(Let
), and we are done.
.
Note:
is the dilation of
's Simson line wrt the medial triangle by a factor of
.
*To be continued*
Let's begin by defining an important point:
Let






Result: If





Before reading on
I want to point out that for the purpose of simplicity, I will only prove the case where
lies inside of
and that
is aute, the other cases can be considered analogously. In this case we can WLOG assume that
are located like so in the diagram. I guess this is one disadvantage of a not-so-insightful proof and it's that you can't generalize.




Diagram

Proof: Let the pedal triangle of





Since













Correlary-2nd Fontene theorem:
If a point moves on a fixed line through the circumcenter, then its pedal circle passes through a fixed point on the nine-point circle.
Proof: Clearly



1st Fontene theorem(Rephrased): The


Proof: Let














Next we will prove that






Note that







Note:



*To be continued*
This post has been edited 7 times. Last edited by XmL, Aug 12, 2014, 11:54 PM