Median triangle

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

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 ${\tfrac {3}{4}}$ 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 ${\tfrac {3}{4}}$.

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.