Let be a

*right triangle*in which , is the foot of the altitude from , is the Kosnita point, and is the symmedian point. Then the lengths of the segments form a geometric progression:

Place the vertices of the right triangle at convenient points: , , and . Then:

- is the point ;
- is the point ;
- is the point .

Find the distance from the Kosnita point to the foot of the altitude from .

Using the given coordinates, we find:

Find the distance from the Kosnita point to the symmedian point.

Using the given coordinates, we find:

Find the distance from vertex to the Kosnita point.

Using the given coordinates, we find:

PROVE that .

Follows from

PROVE that .

Follows from

## Takeaway

In any *right triangle*, the following statements are *equivalent*:

- the right triangle is isosceles
- the Kosnita point coincides with the centroid.

## Task

- (Foot of the symmedian) Let be a
*triangle*having vertices at , , . VERIFY that:- the
*foot of the symmedian from*is - the foot of the symmedian from is equidistant from the feet of the altitudes from and .

- the