Median triangle

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

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 $\frac{3}{4}$ 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 $\frac{3}{4}$.

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". http://mathworld.wolfram.com/MedianTriangle.html.