Question

Напишите задачу на теорему о медианах

Answer 1

В треугольнике ABC медианы AN и СG  взаимно перпендикулярны. AB = c, BC = a. Найти AC

Answer 2

I can understand and communicate fluently in your language of choice. I can also help you with writing, rewriting, improving, or optimizing your content.

You asked me to write a problem on the theorem of medians. Here is one possible problem:

Дан треугольник ABC с медианами AD, BE и CF, пересекающимися в точке G. Известно, что AD = 12 см, BE = 15 см и CF = 18 см. Найдите длины сторон треугольника ABC.

To solve this problem, you can use the following properties of the medians:

- The point G divides each median in the ratio of 2:1, counting from the vertex. This means that AG = 2/3 AD, BG = 2/3 BE and CG = 2/3 CF. - The sum of the squares of the lengths of the medians is equal to three-fourths of the sum of the squares of the lengths of the sides. This means that $$AD^2 + BE^2 + CF^2 = \frac{3}{4} (AB^2 + BC^2 + CA^2)$$

Using these properties, you can find the lengths of the sides by solving a system of three equations. For more information on the theorem of medians, you can check these sources: [Википедия](https://ru.wikipedia.org/wiki/%D0%9C%D0%B5%D0%B4%D0%B8%D0%B0%D0%BD%D0%B0_%D1%82%D1%80%D0%B5%D1%83%D0%B3%D0%BE%D0%BB%D1%8C%D0%BD%D0%B8%D0%BA%D0%B0), [YouTube](https://www.youtube.com/watch?v=qOJLpWFb6jU), [TutorOnline](https://wika.tutoronline.ru/geometriya/class/7/chto-nuzhno-znat-o-mediane-treugolnika), [Wika](https://wika.tutoronline.ru/geometriya/class/7/mediana-v-treugolnikah).