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