Median triangle
From HandWiki
median triangle: [math]\displaystyle{ \triangle BGF }[/math]
reference triangle:[math]\displaystyle{ \triangle ABC }[/math]
median triangle of the median triangle:[math]\displaystyle{ \triangle BKH }[/math]
areas: [math]\displaystyle{ |\triangle BGF|=\tfrac{3}{4}|\triangle ABC| }[/math]
similarity: [math]\displaystyle{ \triangle BGF \sim \triangle BKH }[/math]
ratios: [math]\displaystyle{ \tfrac{|BH|}{|BC|}=\tfrac{|HK|}{|AB|}=\tfrac{|BK|}{|AC|}=\tfrac{3}{4} }[/math]
reference triangle:[math]\displaystyle{ \triangle ABC }[/math]
median triangle of the median triangle:[math]\displaystyle{ \triangle BKH }[/math]
areas: [math]\displaystyle{ |\triangle BGF|=\tfrac{3}{4}|\triangle ABC| }[/math]
similarity: [math]\displaystyle{ \triangle BGF \sim \triangle BKH }[/math]
ratios: [math]\displaystyle{ \tfrac{|BH|}{|BC|}=\tfrac{|HK|}{|AB|}=\tfrac{|BK|}{|AC|}=\tfrac{3}{4} }[/math]
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 [math]\displaystyle{ \tfrac{3}{4} }[/math] 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 [math]\displaystyle{ \tfrac{3}{4} }[/math].
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
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