Question

Дан прямоугольный треугольник АВС, угол С = 90°, СК — высота треугольника АВС, СМ биссектриса треугольника АСК. Докажите, что треугольник ВМСравнобедренный.​

Answer 1

Объяснение:

решение к задаче

Answer 2

I will try to answer your question in detail. Given a right triangle ABC, angle C = 90°, CK is the height of triangle ABC, CM is the bisector of triangle ACK. Prove that triangle BMC is isosceles.

To prove that triangle BMC is isosceles, we need to show that BM = BC, or equivalently, that angle BCM = angle BCM. We can use the following steps:

1. Since CK is the height of triangle ABC, we have that angle ACK = angle BCK = 90°. Therefore, triangle ACK and triangle BCK are right triangles. 2. Since CM is the bisector of angle ACK, we have that angle ACM = angle MCK. Therefore, by the AA criterion, triangle ACM and triangle MCK are similar. 3. From the similarity of triangle ACM and triangle MCK, we have that AC/MC = MC/KC, or equivalently, AC * KC = MC^2. This is the geometric mean theorem for right triangles. 4. Similarly, from the similarity of triangle BCM and triangle MCK, we have that BC/MC = MC/MK, or equivalently, BC * MK = MC^2. This is also the geometric mean theorem for right triangles. 5. From steps 3 and 4, we have that AC * KC = BC * MK, or equivalently, AC/MK = BC/KC. This means that triangle AKM and triangle BKC are similar by the SSS criterion. 6. From the similarity of triangle AKM and triangle BKC, we have that angle AMK = angle BKC. But angle BKC = 90°, so angle AMK = 90° as well. Therefore, angle AMB = angle BMC, since they are both complementary to angle AMK. 7. From step 6, we have that angle AMB = angle BMC. But angle AMB = angle ABC, since they are both opposite to the hypotenuse AC. Therefore, angle ABC = angle BMC, and by the isosceles triangle theorem, triangle BMC is isosceles.

I hope this helps you understand the solution. If you want to see more examples of similar problems, you can check out these links: [Задача 16 геометрия на ЕГЭ-2021 по математике](https://ege-study.ru/ru/ege/podgotovka/matematika/ege-2021-reshenie-zadachi-16/), [Дан прямоугольный треугольник АВС угол С=90 градусов. Точка Е ...](https://class.rambler.ru/temy-geometriya/dan-pryamougolnyy-treugolnik-avs-ugol-s-90-gradusov-tochka-e-prinadlezhit-otrezku-vs-em-perpendikulyaren-avs-dokazhite-chto-as-perpendikulyaren-mv-74213.htm), [Дан прямоугольный треугольник АВС, у которого угол С - прямой, катет ВС ...](https://uchi.ru/otvety/questions/dan-pryamougolniy-treugolnik-avs-u-kotorogo-ugol-s-pryamoy-katet-vs-6-sm-i-ugol-a-60-grad).