The symmedians and the tangential triangle of a triangle | plane geometry | advanced level

Episode 97.

The symmedians and the tangential triangle of a triangle | plane geometry | advanced level.

Branch of mathematics: plane geometry.
Difficulty level: advanced.

Definition. A symmedian $BM'$ of triangle $ABC$ in the line symmetrical to the median $BM$ with respect to the bisector $BD$.

Definition. Let $ABC$ be a triangle and let the tangents to its circumcircle at the vertices $B$ and $C$ intersect at the point $A'$. Similarly, we define the points $B'$ and $C'$. Then $A'B'C'$ is said to be the tangential triangle of the triangle $ABC$.

Theorem. Let $ABC$ be a triangle and let $A'B'C'$ be its tangential triangle. Then the lines $AA'$, $BB'$, $CC'$ are the symmedians of the triangle $ABC$.

Mathematics. Geometry. Plane geometry.
#Mathematics #Geometry #PlaneGeometry

tags:
mathematics,geometry,plane geometry,classical geometry,Euclidean geometry,triangle,line,circle,circumcircle,tangent,tangency point,symmedian,median,bisector,tangential,tangential triangle,homothety,symmetry,axis

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