miércoles, 24 de agosto de 2016

The Outer/Inner Garcia Triangle

In euclidean geometry, the Fuhrmann triangle and the Hexyl triangle (and others) are two special triangles with a plethora of surprising properties. On April, 2014, I introduced two triangles, namely, the Outer Garcia triangle and the Inner Garcia triangle which, as it regards to intersting properties, seem to compete very well with the two triangles cited above. 


Outer-Garcia Triangle: it is the triangle obtained by reflecting the incenter around the midpoints of the reference triangle. This is triangle $A'B'C'$ in the animation.

Inner-Garcia Triangle: it is the triangle obtained by reflecting the vertices of the Outer-Garcia triangle around the sides of the reference triangle. Also, It can be obtained by just reflecting the incenter around the perpendicular bisectors of the reference triangle. This is triangle $A''B''C''$ in the animation.

The constructions and many of the properties associated to these triangles can be found in the Encyclopedia of Triangle Centers, part 4, with notation $X(5587)$.

There are also cubics associated to them which you can find in the website of Bernard Gibert. See, for example, the Spieker-Schiffler Cubic.

martes, 23 de agosto de 2016

Concyclicity Associated to a Radical Axis

Consider two circles with centers $A$ and $B$. Call $A'$, $A''$ the two intersections of the line $AB$ and the circle centered at $A$. Similarly, construct $B'$, $B''$. Call $X$ and $Y$ the intersections of circles $(A)$, $(B)$ with another circle passing through $A''$, $B''$. Let another circle passing through $A''$, $B''$ be intersected by both circles $(A)$, $(B)$ at $Z$, $W$, respectively.

Prove that $XYWZ$ is cyclic.