In our first session, we spent some time on this problem: Let be a triangle in which Let and be the angle bisectors with on and on . Let be the reflection of in the line . Prove that lies on
We discovered a few key properties about this configuration that eventually led to a successful solution:
is cyclic because . The cyclic result only occurs when , and it gives us information about the angles of and consequently those of
is the circumcenter of because and . This result relates with and gives us more angle relations.
is cyclic, which we proved by using the above properties to angle chase . Symmetrically is cyclic, so , which implies are collinear and we are done.
The discovery of these properties makes the problem transparent, meaning we know what makes our main result true. What this also means is that we can probably find simpler solutions. Indeed, there are many ways to solve this problem:
Using angle bisector theorem and Miquel point (Avoids tedious angle chasing):
Since is a cyclic quadrilateral, we know the miqual point of its complete correspondence lies on . This means the circumcircles of all have a common point on , we denote the point . Because is cyclic, ; therefore and similarly . This is enough to establish and lies on
Utilizing as the axis of symmetry:
Let . It suffices to prove bisects . We can achieve this with our first property.
Now I will propose a few follow up questions which you guys should now easily answer:
We keep the notations of our main problem:
is equilateral such that are on opposite sides of . Prove that
Through construct the perpendicular to which intersects at . Prove that
is the circumcenter of , Prove that .