Median triangle

The median triangle of a given (reference) triangle is a triangle, the sides of which are equal and parallel to the medians of its reference triangle. The area of the median triangle is ${\displaystyle {\tfrac {3}{4}}}$ of the area of its reference triangle and the median triangle of the median triangle is similar to reference triangle of the first median triangle with a scaling factor of ${\displaystyle {\tfrac {3}{4}}}$.

median triangle: ${\displaystyle \triangle BGF}$
reference triangle:${\displaystyle \triangle ABC}$
median triangle of the median triangle:${\displaystyle \triangle BKH}$
areas: ${\displaystyle |\triangle BGF|={\tfrac {3}{4}}|\triangle ABC|}$
similarity: ${\displaystyle \triangle BGF\sim \triangle BKH}$
ratios: ${\displaystyle {\tfrac {|BH|}{|BC|}}={\tfrac {|HK|}{|AB|}}={\tfrac {|BK|}{|AC|}}={\tfrac {3}{4}}}$

References

• Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, pp. 282–283
• Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA, 2010, ISBN 9780883853481, p. 165
• Árpad Bényi, Branko Ćurgus: "Outer Median Triangles". In: Mathematics Magazine, Vol. 87, No. 3 (June 2014), pp. 185–194 (JSTOR)

External links

• Weisstein, Eric W. "Median Triangle". MathWorld.